LMFDB - Embedded newform 448.3.d.d.127.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(127,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.d (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:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1539727.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{3} - 8x + 8 $ x^6 - 2x^5 + 3x^3 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.5
Root			$-1.20134 - 0.746179i$ of defining polynomial
Character	$\chi$	$=$	448.127
Dual form			448.3.d.d.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.98472i q^{3} +6.80536 q^{5} +2.64575i q^{7} +0.0914622 q^{9} +O(q^{10})$	$q+2.98472i q^{3} +6.80536 q^{5} +2.64575i q^{7} +0.0914622 q^{9} +14.3426i q^{11} -4.62243 q^{13} +20.3121i q^{15} +11.6107 q^{17} -22.4428i q^{19} -7.89682 q^{21} +0.853941i q^{23} +21.3129 q^{25} +27.1354i q^{27} +42.8321 q^{29} -1.54982i q^{31} -42.8087 q^{33} +18.0053i q^{35} -45.0151 q^{37} -13.7967i q^{39} -36.6492 q^{41} +27.9894i q^{43} +0.622433 q^{45} +42.1740i q^{47} -7.00000 q^{49} +34.6547i q^{51} -43.4044 q^{53} +97.6068i q^{55} +66.9856 q^{57} +53.9979i q^{59} +15.9498 q^{61} +0.241986i q^{63} -31.4573 q^{65} -91.9453i q^{67} -2.54877 q^{69} +16.3585i q^{71} +9.58728 q^{73} +63.6130i q^{75} -37.9470 q^{77} -56.9826i q^{79} -80.1685 q^{81} -14.3077i q^{83} +79.0151 q^{85} +127.842i q^{87} +100.136 q^{89} -12.2298i q^{91} +4.62579 q^{93} -152.732i q^{95} +68.2834 q^{97} +1.31181i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 10 q^{9}+O(q^{10}) $ 6 * q + 4 * q^5 - 10 * q^9	$ 6 q + 4 q^{5} - 10 q^{9} - 12 q^{13} - 4 q^{17} - 30 q^{25} + 36 q^{29} + 80 q^{33} - 28 q^{37} - 20 q^{41} - 12 q^{45} - 42 q^{49} - 92 q^{53} + 160 q^{57} + 164 q^{61} - 136 q^{65} + 48 q^{69} - 132 q^{73} - 112 q^{77} - 218 q^{81} + 232 q^{85} + 348 q^{89} - 288 q^{93} + 252 q^{97}+O(q^{100}) $ 6 * q + 4 * q^5 - 10 * q^9 - 12 * q^13 - 4 * q^17 - 30 * q^25 + 36 * q^29 + 80 * q^33 - 28 * q^37 - 20 * q^41 - 12 * q^45 - 42 * q^49 - 92 * q^53 + 160 * q^57 + 164 * q^61 - 136 * q^65 + 48 * q^69 - 132 * q^73 - 112 * q^77 - 218 * q^81 + 232 * q^85 + 348 * q^89 - 288 * q^93 + 252 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.98472i	0.994906i	0.867491	+	0.497453i	$0.165731\pi$
$3$	2.98472i	0.994906i	−0.867491	+	0.497453i	$0.834269\pi$
$4$	0	0
$5$	6.80536	1.36107	0.680536	−	0.732715i	$-0.261747\pi$
$5$	6.80536	1.36107	0.680536	+	0.732715i	$0.261747\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	0.0914622	0.0101625
$10$	0	0
$11$	14.3426i	1.30388i	0.758272	+	0.651938i	$0.226044\pi$
$11$	14.3426i	1.30388i	−0.758272	+	0.651938i	$0.773956\pi$
$12$	0	0
$13$	−4.62243	−0.355572	−0.177786	−	0.984069i	$-0.556893\pi$
$13$	−4.62243	−0.355572	−0.177786	+	0.984069i	$0.556893\pi$
$14$	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
$19$	− 22.4428i	− 1.18120i	−0.806964	−	0.590601i	$-0.798891\pi$
$19$	− 22.4428i	− 1.18120i	0.806964	−	0.590601i	$-0.201109\pi$
$20$	0	0
$21$	−7.89682	−0.376039
$22$	0	0
$23$	0.853941i	0.0371279i	0.999828	+	0.0185639i	$0.00590943\pi$
$23$	0.853941i	0.0371279i	−0.999828	+	0.0185639i	$0.994091\pi$
$24$	0	0
$25$	21.3129	0.852516
$26$	0	0
$27$	27.1354i	1.00502i
$28$	0	0
$29$	42.8321	1.47697	0.738485	−	0.674270i	$-0.235541\pi$
$29$	42.8321	1.47697	0.738485	+	0.674270i	$0.235541\pi$
$30$	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
$35$	18.0053i	0.514437i
$36$	0	0
$37$	−45.0151	−1.21662	−0.608312	−	0.793698i	$-0.708153\pi$
$37$	−45.0151	−1.21662	−0.608312	+	0.793698i	$0.708153\pi$
$38$	0	0
$39$	− 13.7967i	− 0.353760i
$40$	0	0
$41$	−36.6492	−0.893883	−0.446942	−	0.894563i	$-0.647487\pi$
$41$	−36.6492	−0.893883	−0.446942	+	0.894563i	$0.647487\pi$
$42$	0	0
$43$	27.9894i	0.650916i	0.945557	+	0.325458i	$0.105518\pi$
$43$	27.9894i	0.650916i	−0.945557	+	0.325458i	$0.894482\pi$
$44$	0	0
$45$	0.622433	0.0138318
$46$	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
$51$	34.6547i	0.679504i
$52$	0	0
$53$	−43.4044	−0.818950	−0.409475	−	0.912321i	$-0.634288\pi$
$53$	−43.4044	−0.818950	−0.409475	+	0.912321i	$0.634288\pi$
$54$	0	0
$55$	97.6068i	1.77467i
$56$	0	0
$57$	66.9856	1.17519
$58$	0	0
$59$	53.9979i	0.915219i	0.889153	+	0.457609i	$0.151294\pi$
$59$	53.9979i	0.915219i	−0.889153	+	0.457609i	$0.848706\pi$
$60$	0	0
$61$	15.9498	0.261472	0.130736	−	0.991417i	$-0.458266\pi$
$61$	15.9498	0.261472	0.130736	+	0.991417i	$0.458266\pi$
$62$	0	0
$63$	0.241986i	0.00384105i
$64$	0	0
$65$	−31.4573	−0.483959
$66$	0	0
$67$	− 91.9453i	− 1.37232i	−0.727452	−	0.686159i	$-0.759295\pi$
$67$	− 91.9453i	− 1.37232i	0.727452	−	0.686159i	$-0.240705\pi$
$68$	0	0
$69$	−2.54877	−0.0369387
$70$	0	0
$71$	16.3585i	0.230401i	0.993342	+	0.115201i	$0.0367511\pi$
$71$	16.3585i	0.230401i	−0.993342	+	0.115201i	$0.963249\pi$
$72$	0	0
$73$	9.58728	0.131333	0.0656663	−	0.997842i	$-0.479083\pi$
$73$	9.58728	0.131333	0.0656663	+	0.997842i	$0.479083\pi$
$74$	0	0
$75$	63.6130i	0.848173i
$76$	0	0
$77$	−37.9470	−0.492819
$78$	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
$83$	− 14.3077i	− 0.172381i	−0.996279	−	0.0861907i	$-0.972531\pi$
$83$	− 14.3077i	− 0.172381i	0.996279	−	0.0861907i	$-0.0274694\pi$
$84$	0	0
$85$	79.0151	0.929589
$86$	0	0
$87$	127.842i	1.46945i
$88$	0	0
$89$	100.136	1.12512	0.562562	−	0.826755i	$-0.309816\pi$
$89$	100.136	1.12512	0.562562	+	0.826755i	$0.309816\pi$
$90$	0	0
$91$	− 12.2298i	− 0.134394i
$92$	0	0
$93$	4.62579	0.0497396
$94$	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
$99$	1.31181i	0.0132506i
$100$	0	0
$101$	28.1628	0.278840	0.139420	−	0.990233i	$-0.455476\pi$
$101$	28.1628	0.278840	0.139420	+	0.990233i	$0.455476\pi$
$102$	0	0
$103$	− 160.243i	− 1.55575i	−0.628416	−	0.777877i	$-0.716297\pi$
$103$	− 160.243i	− 1.55575i	0.628416	−	0.777877i	$-0.283703\pi$
$104$	0	0
$105$	−53.7407	−0.511816
$106$	0	0
$107$	− 118.923i	− 1.11143i	−0.831374	−	0.555713i	$-0.812445\pi$
$107$	− 118.923i	− 1.11143i	0.831374	−	0.555713i	$-0.187555\pi$
$108$	0	0
$109$	98.2666	0.901529	0.450764	−	0.892643i	$-0.351151\pi$
$109$	98.2666	0.901529	0.450764	+	0.892643i	$0.351151\pi$
$110$	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
$115$	5.81138i	0.0505337i
$116$	0	0
$117$	−0.422778	−0.00361349
$118$	0	0
$119$	30.7191i	0.258143i
$120$	0	0
$121$	−84.7112	−0.700092
$122$	0	0
$123$	− 109.388i	− 0.889330i
$124$	0	0
$125$	−25.0921	−0.200737
$126$	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
$131$	− 209.837i	− 1.60181i	−0.598791	−	0.800906i	$-0.704352\pi$
$131$	− 209.837i	− 1.60181i	0.598791	−	0.800906i	$-0.295648\pi$
$132$	0	0
$133$	59.3782	0.446453
$134$	0	0
$135$	184.666i	1.36790i
$136$	0	0
$137$	−18.1650	−0.132591	−0.0662954	−	0.997800i	$-0.521118\pi$
$137$	−18.1650	−0.132591	−0.0662954	+	0.997800i	$0.521118\pi$
$138$	0	0
$139$	134.084i	0.964635i	0.875996	+	0.482318i	$0.160205\pi$
$139$	134.084i	0.964635i	−0.875996	+	0.482318i	$0.839795\pi$
$140$	0	0
$141$	−125.877	−0.892747
$142$	0	0
$143$	− 66.2979i	− 0.463621i
$144$	0	0
$145$	291.488	2.01026
$146$	0	0
$147$	− 20.8930i	− 0.142129i
$148$	0	0
$149$	20.7618	0.139341	0.0696706	−	0.997570i	$-0.477805\pi$
$149$	20.7618	0.139341	0.0696706	+	0.997570i	$0.477805\pi$
$150$	0	0
$151$	93.1090i	0.616616i	0.951287	+	0.308308i	$0.0997628\pi$
$151$	93.1090i	0.616616i	−0.951287	+	0.308308i	$0.900237\pi$
$152$	0	0
$153$	1.06194	0.00694080
$154$	0	0
$155$	− 10.5471i	− 0.0680458i
$156$	0	0
$157$	−159.337	−1.01489	−0.507444	−	0.861685i	$-0.669410\pi$
$157$	−159.337	−1.01489	−0.507444	+	0.861685i	$0.669410\pi$
$158$	0	0
$159$	− 129.550i	− 0.814778i
$160$	0	0
$161$	−2.25932	−0.0140330
$162$	0	0
$163$	− 45.8277i	− 0.281152i	−0.990070	−	0.140576i	$-0.955105\pi$
$163$	− 45.8277i	− 0.281152i	0.990070	−	0.140576i	$-0.0448954\pi$
$164$	0	0
$165$	−291.329	−1.76563
$166$	0	0
$167$	− 15.5845i	− 0.0933203i	−0.998911	−	0.0466602i	$-0.985142\pi$
$167$	− 15.5845i	− 0.0933203i	0.998911	−	0.0466602i	$-0.0148578\pi$
$168$	0	0
$169$	−147.633	−0.873569
$170$	0	0
$171$	− 2.05267i	− 0.0120039i
$172$	0	0
$173$	266.021	1.53770	0.768848	−	0.639432i	$-0.220830\pi$
$173$	266.021	1.53770	0.768848	+	0.639432i	$0.220830\pi$
$174$	0	0
$175$	56.3886i	0.322221i
$176$	0	0
$177$	−161.168	−0.910556
$178$	0	0
$179$	− 44.5059i	− 0.248636i	−0.992242	−	0.124318i	$-0.960326\pi$
$179$	− 44.5059i	− 0.248636i	0.992242	−	0.124318i	$-0.0396744\pi$
$180$	0	0
$181$	168.111	0.928787	0.464394	−	0.885629i	$-0.346272\pi$
$181$	168.111	0.928787	0.464394	+	0.885629i	$0.346272\pi$
$182$	0	0
$183$	47.6056i	0.260140i
$184$	0	0
$185$	−306.344	−1.65591
$186$	0	0
$187$	166.528i	0.890525i
$188$	0	0
$189$	−71.7936	−0.379861
$190$	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
$195$	− 93.8912i	− 0.481493i
$196$	0	0
$197$	26.9314	0.136707	0.0683537	−	0.997661i	$-0.478225\pi$
$197$	26.9314	0.136707	0.0683537	+	0.997661i	$0.478225\pi$
$198$	0	0
$199$	− 231.560i	− 1.16362i	−0.813326	−	0.581809i	$-0.802345\pi$
$199$	− 231.560i	− 1.16362i	0.813326	−	0.581809i	$-0.197655\pi$
$200$	0	0
$201$	274.431	1.36533
$202$	0	0
$203$	113.323i	0.558242i
$204$	0	0
$205$	−249.411	−1.21664
$206$	0	0
$207$	0.0781034i	0 0.000377311i
$208$	0	0
$209$	321.890	1.54014
$210$	0	0
$211$	− 79.6903i	− 0.377679i	−0.982008	−	0.188840i	$-0.939527\pi$
$211$	− 79.6903i	− 0.377679i	0.982008	−	0.188840i	$-0.0604726\pi$
$212$	0	0
$213$	−48.8254	−0.229227
$214$	0	0
$215$	190.478i	0.885943i
$216$	0	0
$217$	4.10045	0.0188961
$218$	0	0
$219$	28.6153i	0.130664i
$220$	0	0
$221$	−53.6698	−0.242850
$222$	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
$227$	− 431.324i	− 1.90011i	−0.312088	−	0.950053i	$-0.601028\pi$
$227$	− 431.324i	− 1.90011i	0.312088	−	0.950053i	$-0.398972\pi$
$228$	0	0
$229$	424.901	1.85546	0.927732	−	0.373247i	$-0.121756\pi$
$229$	424.901	1.85546	0.927732	+	0.373247i	$0.121756\pi$
$230$	0	0
$231$	− 113.261i	− 0.490308i
$232$	0	0
$233$	218.828	0.939176	0.469588	−	0.882886i	$-0.344403\pi$
$233$	218.828	0.939176	0.469588	+	0.882886i	$0.344403\pi$
$234$	0	0
$235$	287.009i	1.22131i
$236$	0	0
$237$	170.077	0.717625
$238$	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
$243$	4.93877i	0.0203242i
$244$	0	0
$245$	−47.6375	−0.194439
$246$	0	0
$247$	103.741i	0.420002i
$248$	0	0
$249$	42.7043	0.171503
$250$	0	0
$251$	− 319.525i	− 1.27301i	−0.771274	−	0.636503i	$-0.780380\pi$
$251$	− 319.525i	− 1.27301i	0.771274	−	0.636503i	$-0.219620\pi$
$252$	0	0
$253$	−12.2478	−0.0484101
$254$	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
$259$	− 119.099i	− 0.459840i
$260$	0	0
$261$	3.91752	0.0150097
$262$	0	0
$263$	− 86.1276i	− 0.327481i	−0.986503	−	0.163741i	$-0.947644\pi$
$263$	− 86.1276i	− 0.327481i	0.986503	−	0.163741i	$-0.0523560\pi$
$264$	0	0
$265$	−295.382	−1.11465
$266$	0	0
$267$	298.878i	1.11939i
$268$	0	0
$269$	−269.343	−1.00128	−0.500638	−	0.865657i	$-0.666901\pi$
$269$	−269.343	−1.00128	−0.500638	+	0.865657i	$0.666901\pi$
$270$	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
$275$	305.683i	1.11157i
$276$	0	0
$277$	−356.597	−1.28735	−0.643677	−	0.765298i	$-0.722592\pi$
$277$	−356.597	−1.28735	−0.643677	+	0.765298i	$0.722592\pi$
$278$	0	0
$279$	− 0.141750i	0 0.000508066i
$280$	0	0
$281$	109.846	0.390911	0.195455	−	0.980713i	$-0.437381\pi$
$281$	109.846	0.390911	0.195455	+	0.980713i	$0.437381\pi$
$282$	0	0
$283$	514.582i	1.81831i	0.416457	+	0.909155i	$0.363272\pi$
$283$	514.582i	1.81831i	−0.416457	+	0.909155i	$0.636728\pi$
$284$	0	0
$285$	455.861	1.59951
$286$	0	0
$287$	− 96.9647i	− 0.337856i
$288$	0	0
$289$	−154.191	−0.533534
$290$	0	0
$291$	203.807i	0.700366i
$292$	0	0
$293$	133.002	0.453931	0.226965	−	0.973903i	$-0.427120\pi$
$293$	133.002	0.453931	0.226965	+	0.973903i	$0.427120\pi$
$294$	0	0
$295$	367.475i	1.24568i
$296$	0	0
$297$	−389.194	−1.31042
$298$	0	0
$299$	− 3.94729i	− 0.0132016i
$300$	0	0
$301$	−74.0530	−0.246023
$302$	0	0
$303$	84.0581i	0.277420i
$304$	0	0
$305$	108.544	0.355882
$306$	0	0
$307$	382.811i	1.24694i	0.781847	+	0.623471i	$0.214278\pi$
$307$	382.811i	1.24694i	−0.781847	+	0.623471i	$0.785722\pi$
$308$	0	0
$309$	478.279	1.54783
$310$	0	0
$311$	− 196.374i	− 0.631426i	−0.948855	−	0.315713i	$-0.897756\pi$
$311$	− 196.374i	− 0.631426i	0.948855	−	0.315713i	$-0.102244\pi$
$312$	0	0
$313$	427.045	1.36436	0.682181	−	0.731183i	$-0.261032\pi$
$313$	427.045	1.36436	0.682181	+	0.731183i	$0.261032\pi$
$314$	0	0
$315$	1.64680i	0.00522795i
$316$	0	0
$317$	−111.049	−0.350314	−0.175157	−	0.984541i	$-0.556043\pi$
$317$	−111.049	−0.350314	−0.175157	+	0.984541i	$0.556043\pi$
$318$	0	0
$319$	614.326i	1.92579i
$320$	0	0
$321$	354.951	1.10577
$322$	0	0
$323$	− 260.577i	− 0.806741i
$324$	0	0
$325$	−98.5174	−0.303131
$326$	0	0
$327$	293.298i	0.896936i
$328$	0	0
$329$	−111.582	−0.339154
$330$	0	0
$331$	119.629i	0.361416i	0.983537	+	0.180708i	$0.0578388\pi$
$331$	119.629i	0.361416i	−0.983537	+	0.180708i	$0.942161\pi$
$332$	0	0
$333$	−4.11718	−0.0123639
$334$	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
$339$	289.511i	0.854014i
$340$	0	0
$341$	22.2286	0.0651864
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−17.3453	−0.0502763
$346$	0	0
$347$	268.774i	0.774566i	0.921961	+	0.387283i	$0.126586\pi$
$347$	268.774i	0.774566i	−0.921961	+	0.387283i	$0.873414\pi$
$348$	0	0
$349$	−97.0498	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$349$	−97.0498	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$350$	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
$355$	111.325i	0.313592i
$356$	0	0
$357$	−91.6877	−0.256828
$358$	0	0
$359$	590.194i	1.64399i	0.569492	+	0.821997i	$0.307140\pi$
$359$	590.194i	1.64399i	−0.569492	+	0.821997i	$0.692860\pi$
$360$	0	0
$361$	−142.681	−0.395239
$362$	0	0
$363$	− 252.839i	− 0.696526i
$364$	0	0
$365$	65.2449	0.178753
$366$	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
$371$	− 114.837i	− 0.309534i
$372$	0	0
$373$	−554.016	−1.48530	−0.742649	−	0.669681i	$-0.766431\pi$
$373$	−554.016	−1.48530	−0.742649	+	0.669681i	$0.766431\pi$
$374$	0	0
$375$	− 74.8928i	− 0.199714i
$376$	0	0
$377$	−197.989	−0.525169
$378$	0	0
$379$	− 749.909i	− 1.97865i	−0.145723	−	0.989325i	$-0.546551\pi$
$379$	− 749.909i	− 1.97865i	0.145723	−	0.989325i	$-0.453449\pi$
$380$	0	0
$381$	214.515	0.563032
$382$	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
$387$	2.55997i	0.00661491i
$388$	0	0
$389$	−298.019	−0.766115	−0.383058	−	0.923724i	$-0.625129\pi$
$389$	−298.019	−0.766115	−0.383058	+	0.923724i	$0.625129\pi$
$390$	0	0
$391$	9.91487i	0.0253577i
$392$	0	0
$393$	626.305	1.59365
$394$	0	0
$395$	− 387.787i	− 0.981739i
$396$	0	0
$397$	−127.823	−0.321971	−0.160986	−	0.986957i	$-0.551467\pi$
$397$	−127.823	−0.321971	−0.160986	+	0.986957i	$0.551467\pi$
$398$	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
$403$	7.16396i	0.0177766i
$404$	0	0
$405$	−545.575	−1.34710
$406$	0	0
$407$	− 645.635i	− 1.58633i
$408$	0	0
$409$	666.884	1.63052	0.815262	−	0.579093i	$-0.196593\pi$
$409$	666.884	1.63052	0.815262	+	0.579093i	$0.196593\pi$
$410$	0	0
$411$	− 54.2172i	− 0.131915i
$412$	0	0
$413$	−142.865	−0.345920
$414$	0	0
$415$	− 97.3687i	− 0.234623i
$416$	0	0
$417$	−400.204	−0.959721
$418$	0	0
$419$	95.3634i	0.227598i	0.993504	+	0.113799i	$0.0363019\pi$
$419$	95.3634i	0.227598i	−0.993504	+	0.113799i	$0.963698\pi$
$420$	0	0
$421$	279.158	0.663083	0.331541	−	0.943441i	$-0.392431\pi$
$421$	279.158	0.663083	0.331541	+	0.943441i	$0.392431\pi$
$422$	0	0
$423$	3.85732i	0.00911897i
$424$	0	0
$425$	247.458	0.582254
$426$	0	0
$427$	42.1991i	0.0988270i
$428$	0	0
$429$	197.880	0.461260
$430$	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
$435$	870.010i	2.00002i
$436$	0	0
$437$	19.1649	0.0438555
$438$	0	0
$439$	− 326.773i	− 0.744357i	−0.928161	−	0.372179i	$-0.878611\pi$
$439$	− 326.773i	− 0.744357i	0.928161	−	0.372179i	$-0.121389\pi$
$440$	0	0
$441$	−0.640236	−0.00145178
$442$	0	0
$443$	− 514.789i	− 1.16205i	−0.813885	−	0.581026i	$-0.802651\pi$
$443$	− 514.789i	− 1.16205i	0.813885	−	0.581026i	$-0.197349\pi$
$444$	0	0
$445$	681.462	1.53137
$446$	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
$451$	− 525.646i	− 1.16551i
$452$	0	0
$453$	−277.904	−0.613475
$454$	0	0
$455$	− 83.2282i	− 0.182919i
$456$	0	0
$457$	−264.634	−0.579069	−0.289534	−	0.957168i	$-0.593500\pi$
$457$	−264.634	−0.579069	−0.289534	+	0.957168i	$0.593500\pi$
$458$	0	0
$459$	315.062i	0.686409i
$460$	0	0
$461$	434.788	0.943140	0.471570	−	0.881829i	$-0.343687\pi$
$461$	434.788	0.943140	0.471570	+	0.881829i	$0.343687\pi$
$462$	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
$467$	655.297i	1.40320i	0.712569	+	0.701602i	$0.247532\pi$
$467$	655.297i	1.40320i	−0.712569	+	0.701602i	$0.752468\pi$
$468$	0	0
$469$	243.264	0.518687
$470$	0	0
$471$	− 475.577i	− 1.00972i
$472$	0	0
$473$	−401.442	−0.848714
$474$	0	0
$475$	− 478.322i	− 1.00699i
$476$	0	0
$477$	−3.96986	−0.00832256
$478$	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
$483$	− 6.74342i	− 0.0139615i
$484$	0	0
$485$	464.693	0.958129
$486$	0	0
$487$	− 150.652i	− 0.309347i	−0.987966	−	0.154674i	$-0.950567\pi$
$487$	− 150.652i	− 0.309347i	0.987966	−	0.154674i	$-0.0494326\pi$
$488$	0	0
$489$	136.783	0.279720
$490$	0	0
$491$	− 118.795i	− 0.241946i	−0.992656	−	0.120973i	$-0.961399\pi$
$491$	− 118.795i	− 0.241946i	0.992656	−	0.120973i	$-0.0386014\pi$
$492$	0	0
$493$	497.312	1.00875
$494$	0	0
$495$	8.92733i	0.0180350i
$496$	0	0
$497$	−43.2805	−0.0870834
$498$	0	0
$499$	546.431i	1.09505i	0.836789	+	0.547526i	$0.184430\pi$
$499$	546.431i	1.09505i	−0.836789	+	0.547526i	$0.815570\pi$
$500$	0	0
$501$	46.5153	0.0928449
$502$	0	0
$503$	520.718i	1.03522i	0.855615	+	0.517612i	$0.173179\pi$
$503$	520.718i	1.03522i	−0.855615	+	0.517612i	$0.826821\pi$
$504$	0	0
$505$	191.658	0.379521
$506$	0	0
$507$	− 440.643i	− 0.869119i
$508$	0	0
$509$	−601.427	−1.18159	−0.590793	−	0.806823i	$-0.701185\pi$
$509$	−601.427	−1.18159	−0.590793	+	0.806823i	$0.701185\pi$
$510$	0	0
$511$	25.3656i	0.0496391i
$512$	0	0
$513$	608.997	1.18713
$514$	0	0
$515$	− 1090.51i	− 2.11749i
$516$	0	0
$517$	−604.886	−1.16999
$518$	0	0
$519$	793.998i	1.52986i
$520$	0	0
$521$	−582.268	−1.11760	−0.558798	−	0.829304i	$-0.688737\pi$
$521$	−582.268	−1.11760	−0.558798	+	0.829304i	$0.688737\pi$
$522$	0	0
$523$	143.593i	0.274557i	0.990533	+	0.137278i	$0.0438355\pi$
$523$	143.593i	0.274557i	−0.990533	+	0.137278i	$0.956165\pi$
$524$	0	0
$525$	−168.304	−0.320579
$526$	0	0
$527$	− 17.9946i	− 0.0341453i
$528$	0	0
$529$	528.271	0.998622
$530$	0	0
$531$	4.93877i	0.00930088i
$532$	0	0
$533$	169.409	0.317840
$534$	0	0
$535$	− 809.311i	− 1.51273i
$536$	0	0
$537$	132.838	0.247370
$538$	0	0
$539$	− 100.398i	− 0.186268i
$540$	0	0
$541$	−833.097	−1.53992	−0.769960	−	0.638092i	$-0.779724\pi$
$541$	−833.097	−1.53992	−0.769960	+	0.638092i	$0.779724\pi$
$542$	0	0
$543$	501.762i	0.924056i
$544$	0	0
$545$	668.740	1.22705
$546$	0	0
$547$	356.858i	0.652391i	0.945302	+	0.326196i	$0.105767\pi$
$547$	356.858i	0.652391i	−0.945302	+	0.326196i	$0.894233\pi$
$548$	0	0
$549$	1.45880	0.00265720
$550$	0	0
$551$	− 961.275i	− 1.74460i
$552$	0	0
$553$	150.762	0.272625
$554$	0	0
$555$	− 914.349i	− 1.64748i
$556$	0	0
$557$	−324.859	−0.583230	−0.291615	−	0.956536i	$-0.594193\pi$
$557$	−324.859	−0.583230	−0.291615	+	0.956536i	$0.594193\pi$
$558$	0	0
$559$	− 129.379i	− 0.231447i
$560$	0	0
$561$	−497.040	−0.885989
$562$	0	0
$563$	85.2531i	0.151427i	0.997130	+	0.0757133i	$0.0241234\pi$
$563$	85.2531i	0.151427i	−0.997130	+	0.0757133i	$0.975877\pi$
$564$	0	0
$565$	660.104	1.16833
$566$	0	0
$567$	− 212.106i	− 0.374084i
$568$	0	0
$569$	−28.7512	−0.0505293	−0.0252647	−	0.999681i	$-0.508043\pi$
$569$	−28.7512	−0.0505293	−0.0252647	+	0.999681i	$0.508043\pi$
$570$	0	0
$571$	− 677.532i	− 1.18657i	−0.804992	−	0.593286i	$-0.797830\pi$
$571$	− 677.532i	− 1.18657i	0.804992	−	0.593286i	$-0.202170\pi$
$572$	0	0
$573$	1055.42	1.84191
$574$	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
$579$	− 1040.40i	− 1.79688i
$580$	0	0
$581$	37.8545	0.0651541
$582$	0	0
$583$	− 622.533i	− 1.06781i
$584$	0	0
$585$	−2.87716	−0.00491822
$586$	0	0
$587$	551.489i	0.939504i	0.882798	+	0.469752i	$0.155657\pi$
$587$	551.489i	0.939504i	−0.882798	+	0.469752i	$0.844343\pi$
$588$	0	0
$589$	−34.7825	−0.0590534
$590$	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
$595$	209.054i	0.351352i
$596$	0	0
$597$	691.141	1.15769
$598$	0	0
$599$	− 662.449i	− 1.10593i	−0.833206	−	0.552963i	$-0.813497\pi$
$599$	− 662.449i	− 1.10593i	0.833206	−	0.552963i	$-0.186503\pi$
$600$	0	0
$601$	−840.257	−1.39810	−0.699049	−	0.715074i	$-0.746393\pi$
$601$	−840.257	−1.39810	−0.699049	+	0.715074i	$0.746393\pi$
$602$	0	0
$603$	− 8.40952i	− 0.0139461i
$604$	0	0
$605$	−576.490	−0.952876
$606$	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
$611$	− 194.946i	− 0.319061i
$612$	0	0
$613$	109.148	0.178055	0.0890276	−	0.996029i	$-0.471624\pi$
$613$	109.148	0.178055	0.0890276	+	0.996029i	$0.471624\pi$
$614$	0	0
$615$	− 744.422i	− 1.21044i
$616$	0	0
$617$	78.9177	0.127905	0.0639527	−	0.997953i	$-0.479629\pi$
$617$	78.9177	0.127905	0.0639527	+	0.997953i	$0.479629\pi$
$618$	0	0
$619$	715.995i	1.15670i	0.815790	+	0.578348i	$0.196302\pi$
$619$	715.995i	1.15670i	−0.815790	+	0.578348i	$0.803698\pi$
$620$	0	0
$621$	−23.1721	−0.0373141
$622$	0	0
$623$	264.935i	0.425257i
$624$	0	0
$625$	−703.583	−1.12573
$626$	0	0
$627$	960.749i	1.53230i
$628$	0	0
$629$	−522.657	−0.830933
$630$	0	0
$631$	− 849.316i	− 1.34598i	−0.739650	−	0.672992i	$-0.765009\pi$
$631$	− 849.316i	− 1.34598i	0.739650	−	0.672992i	$-0.234991\pi$
$632$	0	0
$633$	237.853	0.375755
$634$	0	0
$635$	− 489.109i	− 0.770251i
$636$	0	0
$637$	32.3570	0.0507960
$638$	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
$643$	− 257.106i	− 0.399854i	−0.979811	−	0.199927i	$-0.935930\pi$
$643$	− 257.106i	− 0.399854i	0.979811	−	0.199927i	$-0.0640705\pi$
$644$	0	0
$645$	−568.522	−0.881430
$646$	0	0
$647$	699.536i	1.08120i	0.841280	+	0.540600i	$0.181803\pi$
$647$	699.536i	1.08120i	−0.841280	+	0.540600i	$0.818197\pi$
$648$	0	0
$649$	−774.472	−1.19333
$650$	0	0
$651$	12.2387i	0.0187998i
$652$	0	0
$653$	−277.629	−0.425160	−0.212580	−	0.977144i	$-0.568187\pi$
$653$	−277.629	−0.425160	−0.212580	+	0.977144i	$0.568187\pi$
$654$	0	0
$655$	− 1428.02i	− 2.18018i
$656$	0	0
$657$	0.876874	0.00133466
$658$	0	0
$659$	− 522.721i	− 0.793204i	−0.917991	−	0.396602i	$-0.870189\pi$
$659$	− 522.721i	− 0.793204i	0.917991	−	0.396602i	$-0.129811\pi$
$660$	0	0
$661$	−515.922	−0.780518	−0.390259	−	0.920705i	$-0.627615\pi$
$661$	−515.922	−0.780518	−0.390259	+	0.920705i	$0.627615\pi$
$662$	0	0
$663$	− 160.189i	− 0.241612i
$664$	0	0
$665$	404.090	0.607654
$666$	0	0
$667$	36.5761i	0.0548368i
$668$	0	0
$669$	−789.060	−1.17946
$670$	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
$675$	578.335i	0.856792i
$676$	0	0
$677$	576.563	0.851645	0.425822	−	0.904807i	$-0.359985\pi$
$677$	576.563	0.851645	0.425822	+	0.904807i	$0.359985\pi$
$678$	0	0
$679$	180.661i	0.266069i
$680$	0	0
$681$	1287.38	1.89043
$682$	0	0
$683$	− 161.395i	− 0.236302i	−0.992996	−	0.118151i	$-0.962303\pi$
$683$	− 161.395i	− 0.236302i	0.992996	−	0.118151i	$-0.0376968\pi$
$684$	0	0
$685$	−123.619	−0.180466
$686$	0	0
$687$	1268.21i	1.84601i
$688$	0	0
$689$	200.634	0.291196
$690$	0	0
$691$	− 113.661i	− 0.164488i	−0.996612	−	0.0822442i	$-0.973791\pi$
$691$	− 113.661i	− 0.164488i	0.996612	−	0.0822442i	$-0.0262087\pi$
$692$	0	0
$693$	−3.47072	−0.00500826
$694$	0	0
$695$	912.491i	1.31294i
$696$	0	0
$697$	−425.524	−0.610507
$698$	0	0
$699$	653.140i	0.934391i
$700$	0	0
$701$	−331.011	−0.472198	−0.236099	−	0.971729i	$-0.575869\pi$
$701$	−331.011	−0.472198	−0.236099	+	0.971729i	$0.575869\pi$
$702$	0	0
$703$	1010.27i	1.43708i
$704$	0	0
$705$	−856.640	−1.21509
$706$	0	0
$707$	74.5119i	0.105392i
$708$	0	0
$709$	591.351	0.834063	0.417032	−	0.908892i	$-0.363070\pi$
$709$	591.351	0.834063	0.417032	+	0.908892i	$0.363070\pi$
$710$	0	0
$711$	− 5.21176i	− 0.00733018i
$712$	0	0
$713$	1.32346	0.00185618
$714$	0	0
$715$	− 451.181i	− 0.631022i
$716$	0	0
$717$	269.119	0.375340
$718$	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
$723$	1122.09i	1.55199i
$724$	0	0
$725$	912.877	1.25914
$726$	0	0
$727$	659.508i	0.907163i	0.891215	+	0.453582i	$0.149854\pi$
$727$	659.508i	0.907163i	−0.891215	+	0.453582i	$0.850146\pi$
$728$	0	0
$729$	−736.257	−1.00995
$730$	0	0
$731$	324.977i	0.444565i
$732$	0	0
$733$	−211.776	−0.288916	−0.144458	−	0.989511i	$-0.546144\pi$
$733$	−211.776	−0.288916	−0.144458	+	0.989511i	$0.546144\pi$
$734$	0	0
$735$	− 142.184i	− 0.193448i
$736$	0	0
$737$	1318.74	1.78933
$738$	0	0
$739$	246.912i	0.334116i	0.985947	+	0.167058i	$0.0534267\pi$
$739$	246.912i	0.334116i	−0.985947	+	0.167058i	$0.946573\pi$
$740$	0	0
$741$	−309.636	−0.417863
$742$	0	0
$743$	− 333.126i	− 0.448352i	−0.974549	−	0.224176i	$-0.928031\pi$
$743$	− 333.126i	− 0.448352i	0.974549	−	0.224176i	$-0.0719691\pi$
$744$	0	0
$745$	141.292	0.189653
$746$	0	0
$747$	− 1.30861i	− 0.00175182i
$748$	0	0
$749$	314.640	0.420080
$750$	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
$755$	633.640i	0.839259i
$756$	0	0
$757$	−439.344	−0.580375	−0.290187	−	0.956970i	$-0.593718\pi$
$757$	−439.344	−0.580375	−0.290187	+	0.956970i	$0.593718\pi$
$758$	0	0
$759$	− 36.5561i	− 0.0481635i
$760$	0	0
$761$	−460.968	−0.605740	−0.302870	−	0.953032i	$-0.597945\pi$
$761$	−460.968	−0.605740	−0.302870	+	0.953032i	$0.597945\pi$
$762$	0	0
$763$	259.989i	0.340746i
$764$	0	0
$765$	7.22689	0.00944692
$766$	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
$771$	− 499.838i	− 0.648299i
$772$	0	0
$773$	−245.480	−0.317568	−0.158784	−	0.987313i	$-0.550757\pi$
$773$	−245.480	−0.317568	−0.158784	+	0.987313i	$0.550757\pi$
$774$	0	0
$775$	− 33.0312i	− 0.0426209i
$776$	0	0
$777$	355.476	0.457498
$778$	0	0
$779$	822.513i	1.05586i
$780$	0	0
$781$	−234.624	−0.300414
$782$	0	0
$783$	1162.27i	1.48438i
$784$	0	0
$785$	−1084.35	−1.38134
$786$	0	0
$787$	579.993i	0.736967i	0.929634	+	0.368483i	$0.120123\pi$
$787$	579.993i	0.736967i	−0.929634	+	0.368483i	$0.879877\pi$
$788$	0	0
$789$	257.067	0.325813
$790$	0	0
$791$	256.632i	0.324440i
$792$	0	0
$793$	−73.7268	−0.0929720
$794$	0	0
$795$	− 881.632i	− 1.10897i
$796$	0	0
$797$	1348.22	1.69162	0.845810	−	0.533484i	$-0.179118\pi$
$797$	1348.22	1.69162	0.845810	+	0.533484i	$0.179118\pi$
$798$	0	0
$799$	489.670i	0.612853i
$800$	0	0
$801$	9.15867	0.0114340
$802$	0	0
$803$	137.507i	0.171241i
$804$	0	0
$805$	−15.3755	−0.0190999
$806$	0	0
$807$	− 803.913i	− 0.996175i
$808$	0	0
$809$	−697.465	−0.862132	−0.431066	−	0.902320i	$-0.641862\pi$
$809$	−697.465	−0.862132	−0.431066	+	0.902320i	$0.641862\pi$
$810$	0	0
$811$	− 482.271i	− 0.594662i	−0.954774	−	0.297331i	$-0.903903\pi$
$811$	− 482.271i	− 0.594662i	0.954774	−	0.297331i	$-0.0960965\pi$
$812$	0	0
$813$	−950.089	−1.16862
$814$	0	0
$815$	− 311.874i	− 0.382668i
$816$	0	0
$817$	628.161	0.768863
$818$	0	0
$819$	− 1.11857i	− 0.00136577i
$820$	0	0
$821$	386.345	0.470579	0.235289	−	0.971925i	$-0.424396\pi$
$821$	386.345	0.470579	0.235289	+	0.971925i	$0.424396\pi$
$822$	0	0
$823$	354.386i	0.430603i	0.976548	+	0.215301i	$0.0690734\pi$
$823$	354.386i	0.430603i	−0.976548	+	0.215301i	$0.930927\pi$
$824$	0	0
$825$	−912.377	−1.10591
$826$	0	0
$827$	1260.47i	1.52414i	0.647493	+	0.762072i	$0.275818\pi$
$827$	1260.47i	1.52414i	−0.647493	+	0.762072i	$0.724182\pi$
$828$	0	0
$829$	1443.69	1.74148	0.870739	−	0.491746i	$-0.163641\pi$
$829$	1443.69	1.74148	0.870739	+	0.491746i	$0.163641\pi$
$830$	0	0
$831$	− 1064.34i	− 1.28080i
$832$	0	0
$833$	−81.2750	−0.0975690
$834$	0	0
$835$	− 106.058i	− 0.127016i
$836$	0	0
$837$	42.0552	0.0502451
$838$	0	0
$839$	44.6511i	0.0532194i	0.999646	+	0.0266097i	$0.00847114\pi$
$839$	44.6511i	0.0532194i	−0.999646	+	0.0266097i	$0.991529\pi$
$840$	0	0
$841$	993.593	1.18144
$842$	0	0
$843$	327.859i	0.388920i
$844$	0	0
$845$	−1004.70	−1.18899
$846$	0	0
$847$	− 224.125i	− 0.264610i
$848$	0	0
$849$	−1535.88	−1.80905
$850$	0	0
$851$	− 38.4402i	− 0.0451707i
$852$	0	0
$853$	118.167	0.138531	0.0692655	−	0.997598i	$-0.477934\pi$
$853$	118.167	0.138531	0.0692655	+	0.997598i	$0.477934\pi$
$854$	0	0
$855$	− 13.9692i	− 0.0163382i
$856$	0	0
$857$	−153.130	−0.178682	−0.0893408	−	0.996001i	$-0.528476\pi$
$857$	−153.130	−0.178682	−0.0893408	+	0.996001i	$0.528476\pi$
$858$	0	0
$859$	136.656i	0.159087i	0.996831	+	0.0795437i	$0.0253463\pi$
$859$	136.656i	0.159087i	−0.996831	+	0.0795437i	$0.974654\pi$
$860$	0	0
$861$	289.412	0.336135
$862$	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
$867$	− 460.217i	− 0.530816i
$868$	0	0
$869$	817.281	0.940484
$870$	0	0
$871$	425.011i	0.487957i
$872$	0	0
$873$	6.24535	0.00715389
$874$	0	0
$875$	− 66.3874i	− 0.0758713i
$876$	0	0
$877$	−29.8881	−0.0340799	−0.0170399	−	0.999855i	$-0.505424\pi$
$877$	−29.8881	−0.0340799	−0.0170399	+	0.999855i	$0.505424\pi$
$878$	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
$883$	1430.98i	1.62059i	0.586022	+	0.810295i	$0.300693\pi$
$883$	1430.98i	1.62059i	−0.586022	+	0.810295i	$0.699307\pi$
$884$	0	0
$885$	−1096.81	−1.23933
$886$	0	0
$887$	1187.21i	1.33846i	0.743057	+	0.669228i	$0.233375\pi$
$887$	1187.21i	1.33846i	−0.743057	+	0.669228i	$0.766625\pi$
$888$	0	0
$889$	190.153	0.213896
$890$	0	0
$891$	− 1149.83i	− 1.29049i
$892$	0	0
$893$	946.504	1.05991
$894$	0	0
$895$	− 302.879i	− 0.338412i
$896$	0	0
$897$	11.7815	0.0131344
$898$	0	0
$899$	− 66.3823i	− 0.0738401i
$900$	0	0
$901$	−503.956	−0.559329
$902$	0	0
$903$	− 221.027i	− 0.244770i
$904$	0	0
$905$	1144.05	1.26415
$906$	0	0
$907$	− 818.659i	− 0.902601i	−0.892372	−	0.451301i	$-0.850960\pi$
$907$	− 818.659i	− 0.902601i	0.892372	−	0.451301i	$-0.149040\pi$
$908$	0	0
$909$	2.57584	0.00283370
$910$	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
$915$	323.973i	0.354069i
$916$	0	0
$917$	555.177	0.605428
$918$	0	0
$919$	1695.99i	1.84547i	0.385437	+	0.922734i	$0.374051\pi$
$919$	1695.99i	1.84547i	−0.385437	+	0.922734i	$0.625949\pi$
$920$	0	0
$921$	−1142.58	−1.24059
$922$	0	0
$923$	− 75.6160i	− 0.0819241i
$924$	0	0
$925$	−959.401	−1.03719
$926$	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
$931$	157.100i	0.168743i
$932$	0	0
$933$	586.120	0.628210
$934$	0	0
$935$	1133.28i	1.21207i
$936$	0	0
$937$	−315.505	−0.336719	−0.168359	−	0.985726i	$-0.553847\pi$
$937$	−315.505	−0.336719	−0.168359	+	0.985726i	$0.553847\pi$
$938$	0	0
$939$	1274.61i	1.35741i
$940$	0	0
$941$	423.505	0.450058	0.225029	−	0.974352i	$-0.427752\pi$
$941$	423.505	0.450058	0.225029	+	0.974352i	$0.427752\pi$
$942$	0	0
$943$	− 31.2963i	− 0.0331880i
$944$	0	0
$945$	−488.581	−0.517017
$946$	0	0
$947$	− 1147.28i	− 1.21149i	−0.795660	−	0.605743i	$-0.792876\pi$
$947$	− 1147.28i	− 1.21149i	0.795660	−	0.605743i	$-0.207124\pi$
$948$	0	0
$949$	−44.3166	−0.0466982
$950$	0	0
$951$	− 331.451i	− 0.348529i
$952$	0	0
$953$	736.494	0.772816	0.386408	−	0.922328i	$-0.373716\pi$
$953$	736.494	0.772816	0.386408	+	0.922328i	$0.373716\pi$
$954$	0	0
$955$	− 2406.42i	− 2.51981i
$956$	0	0
$957$	−1833.59	−1.91598
$958$	0	0
$959$	− 48.0599i	− 0.0501146i
$960$	0	0
$961$	958.598	0.997501
$962$	0	0
$963$	− 10.8769i	− 0.0112948i
$964$	0	0
$965$	−2372.17	−2.45821
$966$	0	0
$967$	1147.66i	1.18682i	0.804899	+	0.593411i	$0.202219\pi$
$967$	1147.66i	1.18682i	−0.804899	+	0.593411i	$0.797781\pi$
$968$	0	0
$969$	777.750	0.802632
$970$	0	0
$971$	1108.64i	1.14175i	0.821036	+	0.570876i	$0.193396\pi$
$971$	1108.64i	1.14175i	−0.821036	+	0.570876i	$0.806604\pi$
$972$	0	0
$973$	−354.754	−0.364598
$974$	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
$979$	1436.21i	1.46702i
$980$	0	0
$981$	8.98769	0.00916176
$982$	0	0
$983$	− 79.5940i	− 0.0809705i	−0.999180	−	0.0404853i	$-0.987110\pi$
$983$	− 79.5940i	− 0.0809705i	0.999180	−	0.0404853i	$-0.0128904\pi$
$984$	0	0
$985$	183.277	0.186069
$986$	0	0
$987$	− 333.040i	− 0.337427i
$988$	0	0
$989$	−23.9013	−0.0241671
$990$	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
$995$	− 1575.85i	− 1.58377i
$996$	0	0
$997$	1739.99	1.74523	0.872615	−	0.488409i	$-0.162422\pi$
$997$	1739.99	1.74523	0.872615	+	0.488409i	$0.162422\pi$
$998$	0	0
$999$	− 1221.50i	− 1.22273i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.d.d.127.5		6
4.3	odd	2	inner	448.3.d.d.127.2		6
8.3	odd	2		28.3.c.a.15.5	✓	6
8.5	even	2		28.3.c.a.15.6	yes	6
16.3	odd	4		1792.3.g.g.127.9		12
16.5	even	4		1792.3.g.g.127.10		12
16.11	odd	4		1792.3.g.g.127.4		12
16.13	even	4		1792.3.g.g.127.3		12
24.5	odd	2		252.3.g.a.127.1		6
24.11	even	2		252.3.g.a.127.2		6
56.3	even	6		196.3.g.j.79.3		12
56.5	odd	6		196.3.g.j.67.3		12
56.11	odd	6		196.3.g.k.79.3		12
56.13	odd	2		196.3.c.g.99.6		6
56.19	even	6		196.3.g.j.67.1		12
56.27	even	2		196.3.c.g.99.5		6
56.37	even	6		196.3.g.k.67.3		12
56.45	odd	6		196.3.g.j.79.1		12
56.51	odd	6		196.3.g.k.67.1		12
56.53	even	6		196.3.g.k.79.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.c.a.15.5	✓	6	8.3	odd	2
28.3.c.a.15.6	yes	6	8.5	even	2
196.3.c.g.99.5		6	56.27	even	2
196.3.c.g.99.6		6	56.13	odd	2
196.3.g.j.67.1		12	56.19	even	6
196.3.g.j.67.3		12	56.5	odd	6
196.3.g.j.79.1		12	56.45	odd	6
196.3.g.j.79.3		12	56.3	even	6
196.3.g.k.67.1		12	56.51	odd	6
196.3.g.k.67.3		12	56.37	even	6
196.3.g.k.79.1		12	56.53	even	6
196.3.g.k.79.3		12	56.11	odd	6
252.3.g.a.127.1		6	24.5	odd	2
252.3.g.a.127.2		6	24.11	even	2
448.3.d.d.127.2		6	4.3	odd	2	inner
448.3.d.d.127.5		6	1.1	even	1	trivial
1792.3.g.g.127.3		12	16.13	even	4
1792.3.g.g.127.4		12	16.11	odd	4
1792.3.g.g.127.9		12	16.3	odd	4
1792.3.g.g.127.10		12	16.5	even	4

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.98472i q^{3} +6.80536 q^{5} +2.64575i q^{7} +0.0914622 q^{9} +O(q^{10})\)	\(q+2.98472i q^{3} +6.80536 q^{5} +2.64575i q^{7} +0.0914622 q^{9} +14.3426i q^{11} -4.62243 q^{13} +20.3121i q^{15} +11.6107 q^{17} -22.4428i q^{19} -7.89682 q^{21} +0.853941i q^{23} +21.3129 q^{25} +27.1354i q^{27} +42.8321 q^{29} -1.54982i q^{31} -42.8087 q^{33} +18.0053i q^{35} -45.0151 q^{37} -13.7967i q^{39} -36.6492 q^{41} +27.9894i q^{43} +0.622433 q^{45} +42.1740i q^{47} -7.00000 q^{49} +34.6547i q^{51} -43.4044 q^{53} +97.6068i q^{55} +66.9856 q^{57} +53.9979i q^{59} +15.9498 q^{61} +0.241986i q^{63} -31.4573 q^{65} -91.9453i q^{67} -2.54877 q^{69} +16.3585i q^{71} +9.58728 q^{73} +63.6130i q^{75} -37.9470 q^{77} -56.9826i q^{79} -80.1685 q^{81} -14.3077i q^{83} +79.0151 q^{85} +127.842i q^{87} +100.136 q^{89} -12.2298i q^{91} +4.62579 q^{93} -152.732i q^{95} +68.2834 q^{97} +1.31181i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 10 q^{9}+O(q^{10}) \) 6 * q + 4 * q^5 - 10 * q^9	\( 6 q + 4 q^{5} - 10 q^{9} - 12 q^{13} - 4 q^{17} - 30 q^{25} + 36 q^{29} + 80 q^{33} - 28 q^{37} - 20 q^{41} - 12 q^{45} - 42 q^{49} - 92 q^{53} + 160 q^{57} + 164 q^{61} - 136 q^{65} + 48 q^{69} - 132 q^{73} - 112 q^{77} - 218 q^{81} + 232 q^{85} + 348 q^{89} - 288 q^{93} + 252 q^{97}+O(q^{100}) \) 6 * q + 4 * q^5 - 10 * q^9 - 12 * q^13 - 4 * q^17 - 30 * q^25 + 36 * q^29 + 80 * q^33 - 28 * q^37 - 20 * q^41 - 12 * q^45 - 42 * q^49 - 92 * q^53 + 160 * q^57 + 164 * q^61 - 136 * q^65 + 48 * q^69 - 132 * q^73 - 112 * q^77 - 218 * q^81 + 232 * q^85 + 348 * q^89 - 288 * q^93 + 252 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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