LMFDB - Embedded newform 480.3.g.a.271.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,3,Mod(271,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.271"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	480.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.0790526893$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + x^{14} + 24 x^{13} - 44 x^{12} - 32 x^{11} + 180 x^{10} - 64 x^{9} - 352 x^{8} + \cdots + 65536 $ x^16 - 4x^15 + x^14 + 24x^13 - 44x^12 - 32x^11 + 180x^10 - 64x^9 - 352x^8 - 256x^7 + 2880x^6 - 2048x^5 - 11264x^4 + 24576x^3 + 4096x^2 - 65536x + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.11
Root			$0.444906 + 1.94989i$ of defining polynomial
Character	$\chi$	$=$	480.271
Dual form			480.3.g.a.271.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -2.23607i q^{5} +5.48395i q^{7} +3.00000 q^{9} -1.62564 q^{11} -18.0504i q^{13} -3.87298i q^{15} +24.0180 q^{17} +34.4282 q^{19} +9.49847i q^{21} +30.7256i q^{23} -5.00000 q^{25} +5.19615 q^{27} -8.33039i q^{29} -5.48491i q^{31} -2.81570 q^{33} +12.2625 q^{35} -42.8951i q^{37} -31.2641i q^{39} -6.22670 q^{41} +57.2727 q^{43} -6.70820i q^{45} -41.5525i q^{47} +18.9263 q^{49} +41.6005 q^{51} +38.1670i q^{53} +3.63505i q^{55} +59.6314 q^{57} -70.7342 q^{59} +63.5799i q^{61} +16.4518i q^{63} -40.3618 q^{65} -59.4604 q^{67} +53.2182i q^{69} -48.4629i q^{71} +93.1375 q^{73} -8.66025 q^{75} -8.91495i q^{77} +60.2424i q^{79} +9.00000 q^{81} -5.69214 q^{83} -53.7060i q^{85} -14.4287i q^{87} +49.1046 q^{89} +98.9872 q^{91} -9.50015i q^{93} -76.9838i q^{95} +35.9229 q^{97} -4.87693 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9} - 64 q^{11} + 32 q^{19} - 80 q^{25} + 192 q^{43} - 80 q^{49} - 96 q^{51} + 96 q^{57} - 128 q^{59} + 64 q^{67} - 160 q^{73} + 144 q^{81} + 192 q^{91} - 224 q^{97} - 192 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 64 * q^11 + 32 * q^19 - 80 * q^25 + 192 * q^43 - 80 * q^49 - 96 * q^51 + 96 * q^57 - 128 * q^59 + 64 * q^67 - 160 * q^73 + 144 * q^81 + 192 * q^91 - 224 * q^97 - 192 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	5.48395i	0.783421i	0.920089	+	0.391710i	$0.128116\pi$
$7$	5.48395i	0.783421i	−0.920089	+	0.391710i	$0.871884\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−1.62564	−0.147786	−0.0738929	−	0.997266i	$-0.523542\pi$
$11$	−1.62564	−0.147786	−0.0738929	+	0.997266i	$0.523542\pi$
$12$	0	0
$13$	− 18.0504i	− 1.38849i	−0.719739	−	0.694245i	$-0.755739\pi$
$13$	− 18.0504i	− 1.38849i	0.719739	−	0.694245i	$-0.244261\pi$
$14$	0	0
$15$	− 3.87298i	− 0.258199i
$16$	0	0
$17$	24.0180	1.41283	0.706413	−	0.707800i	$-0.250312\pi$
$17$	24.0180	1.41283	0.706413	+	0.707800i	$0.250312\pi$
$18$	0	0
$19$	34.4282	1.81201	0.906005	−	0.423267i	$-0.139117\pi$
$19$	34.4282	1.81201	0.906005	+	0.423267i	$0.139117\pi$
$20$	0	0
$21$	9.49847i	0.452308i
$22$	0	0
$23$	30.7256i	1.33589i	0.744209	+	0.667947i	$0.232827\pi$
$23$	30.7256i	1.33589i	−0.744209	+	0.667947i	$0.767173\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	− 8.33039i	− 0.287255i	−0.989632	−	0.143627i	$-0.954123\pi$
$29$	− 8.33039i	− 0.287255i	0.989632	−	0.143627i	$-0.0458767\pi$
$30$	0	0
$31$	− 5.48491i	− 0.176933i	−0.996079	−	0.0884663i	$-0.971803\pi$
$31$	− 5.48491i	− 0.176933i	0.996079	−	0.0884663i	$-0.0281966\pi$
$32$	0	0
$33$	−2.81570	−0.0853242
$34$	0	0
$35$	12.2625	0.350356
$36$	0	0
$37$	− 42.8951i	− 1.15933i	−0.814856	−	0.579664i	$-0.803184\pi$
$37$	− 42.8951i	− 1.15933i	0.814856	−	0.579664i	$-0.196816\pi$
$38$	0	0
$39$	− 31.2641i	− 0.801645i
$40$	0	0
$41$	−6.22670	−0.151871	−0.0759354	−	0.997113i	$-0.524194\pi$
$41$	−6.22670	−0.151871	−0.0759354	+	0.997113i	$0.524194\pi$
$42$	0	0
$43$	57.2727	1.33192	0.665961	−	0.745986i	$-0.268022\pi$
$43$	57.2727	1.33192	0.665961	+	0.745986i	$0.268022\pi$
$44$	0	0
$45$	− 6.70820i	− 0.149071i
$46$	0	0
$47$	− 41.5525i	− 0.884095i	−0.896992	−	0.442048i	$-0.854252\pi$
$47$	− 41.5525i	− 0.884095i	0.896992	−	0.442048i	$-0.145748\pi$
$48$	0	0
$49$	18.9263	0.386252
$50$	0	0
$51$	41.6005	0.815695
$52$	0	0
$53$	38.1670i	0.720132i	0.932927	+	0.360066i	$0.117246\pi$
$53$	38.1670i	0.720132i	−0.932927	+	0.360066i	$0.882754\pi$
$54$	0	0
$55$	3.63505i	0.0660918i
$56$	0	0
$57$	59.6314	1.04616
$58$	0	0
$59$	−70.7342	−1.19889	−0.599443	−	0.800418i	$-0.704611\pi$
$59$	−70.7342	−1.19889	−0.599443	+	0.800418i	$0.704611\pi$
$60$	0	0
$61$	63.5799i	1.04229i	0.853467	+	0.521147i	$0.174496\pi$
$61$	63.5799i	1.04229i	−0.853467	+	0.521147i	$0.825504\pi$
$62$	0	0
$63$	16.4518i	0.261140i
$64$	0	0
$65$	−40.3618	−0.620951
$66$	0	0
$67$	−59.4604	−0.887468	−0.443734	−	0.896159i	$-0.646346\pi$
$67$	−59.4604	−0.887468	−0.443734	+	0.896159i	$0.646346\pi$
$68$	0	0
$69$	53.2182i	0.771278i
$70$	0	0
$71$	− 48.4629i	− 0.682576i	−0.939959	−	0.341288i	$-0.889137\pi$
$71$	− 48.4629i	− 0.682576i	0.939959	−	0.341288i	$-0.110863\pi$
$72$	0	0
$73$	93.1375	1.27586	0.637928	−	0.770096i	$-0.279792\pi$
$73$	93.1375	1.27586	0.637928	+	0.770096i	$0.279792\pi$
$74$	0	0
$75$	−8.66025	−0.115470
$76$	0	0
$77$	− 8.91495i	− 0.115779i
$78$	0	0
$79$	60.2424i	0.762562i	0.924459	+	0.381281i	$0.124517\pi$
$79$	60.2424i	0.762562i	−0.924459	+	0.381281i	$0.875483\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−5.69214	−0.0685800	−0.0342900	−	0.999412i	$-0.510917\pi$
$83$	−5.69214	−0.0685800	−0.0342900	+	0.999412i	$0.510917\pi$
$84$	0	0
$85$	− 53.7060i	− 0.631835i
$86$	0	0
$87$	− 14.4287i	− 0.165847i
$88$	0	0
$89$	49.1046	0.551737	0.275868	−	0.961195i	$-0.411035\pi$
$89$	49.1046	0.551737	0.275868	+	0.961195i	$0.411035\pi$
$90$	0	0
$91$	98.9872	1.08777
$92$	0	0
$93$	− 9.50015i	− 0.102152i
$94$	0	0
$95$	− 76.9838i	− 0.810356i
$96$	0	0
$97$	35.9229	0.370339	0.185170	−	0.982707i	$-0.440717\pi$
$97$	35.9229	0.370339	0.185170	+	0.982707i	$0.440717\pi$
$98$	0	0
$99$	−4.87693	−0.0492620
$100$	0	0
$101$	67.3302i	0.666635i	0.942815	+	0.333318i	$0.108168\pi$
$101$	67.3302i	0.666635i	−0.942815	+	0.333318i	$0.891832\pi$
$102$	0	0
$103$	73.3368i	0.712008i	0.934484	+	0.356004i	$0.115861\pi$
$103$	73.3368i	0.712008i	−0.934484	+	0.356004i	$0.884139\pi$
$104$	0	0
$105$	21.2392	0.202278
$106$	0	0
$107$	−154.194	−1.44107	−0.720533	−	0.693421i	$-0.756103\pi$
$107$	−154.194	−1.44107	−0.720533	+	0.693421i	$0.756103\pi$
$108$	0	0
$109$	− 120.753i	− 1.10782i	−0.832575	−	0.553912i	$-0.813134\pi$
$109$	− 120.753i	− 1.10782i	0.832575	−	0.553912i	$-0.186866\pi$
$110$	0	0
$111$	− 74.2965i	− 0.669338i
$112$	0	0
$113$	−209.411	−1.85320	−0.926599	−	0.376050i	$-0.877282\pi$
$113$	−209.411	−1.85320	−0.926599	+	0.376050i	$0.877282\pi$
$114$	0	0
$115$	68.7044	0.597430
$116$	0	0
$117$	− 54.1511i	− 0.462830i
$118$	0	0
$119$	131.714i	1.10684i
$120$	0	0
$121$	−118.357	−0.978159
$122$	0	0
$123$	−10.7850	−0.0876827
$124$	0	0
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	− 137.663i	− 1.08396i	−0.840391	−	0.541981i	$-0.817674\pi$
$127$	− 137.663i	− 1.08396i	0.840391	−	0.541981i	$-0.182326\pi$
$128$	0	0
$129$	99.1992	0.768986
$130$	0	0
$131$	−30.3019	−0.231312	−0.115656	−	0.993289i	$-0.536897\pi$
$131$	−30.3019	−0.231312	−0.115656	+	0.993289i	$0.536897\pi$
$132$	0	0
$133$	188.802i	1.41957i
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	−186.067	−1.35815	−0.679076	−	0.734068i	$-0.737619\pi$
$137$	−186.067	−1.35815	−0.679076	+	0.734068i	$0.737619\pi$
$138$	0	0
$139$	−5.92262	−0.0426087	−0.0213044	−	0.999773i	$-0.506782\pi$
$139$	−5.92262	−0.0426087	−0.0213044	+	0.999773i	$0.506782\pi$
$140$	0	0
$141$	− 71.9710i	− 0.510433i
$142$	0	0
$143$	29.3435i	0.205199i
$144$	0	0
$145$	−18.6273	−0.128464
$146$	0	0
$147$	32.7814	0.223003
$148$	0	0
$149$	235.886i	1.58313i	0.611086	+	0.791564i	$0.290733\pi$
$149$	235.886i	1.58313i	−0.611086	+	0.791564i	$0.709267\pi$
$150$	0	0
$151$	56.9382i	0.377074i	0.982066	+	0.188537i	$0.0603745\pi$
$151$	56.9382i	0.377074i	−0.982066	+	0.188537i	$0.939625\pi$
$152$	0	0
$153$	72.0541	0.470942
$154$	0	0
$155$	−12.2646	−0.0791267
$156$	0	0
$157$	− 304.162i	− 1.93734i	−0.248351	−	0.968670i	$-0.579889\pi$
$157$	− 304.162i	− 1.93734i	0.248351	−	0.968670i	$-0.420111\pi$
$158$	0	0
$159$	66.1072i	0.415769i
$160$	0	0
$161$	−168.497	−1.04657
$162$	0	0
$163$	−248.671	−1.52559	−0.762796	−	0.646640i	$-0.776174\pi$
$163$	−248.671	−1.52559	−0.762796	+	0.646640i	$0.776174\pi$
$164$	0	0
$165$	6.29609i	0.0381581i
$166$	0	0
$167$	20.4759i	0.122610i	0.998119	+	0.0613051i	$0.0195263\pi$
$167$	20.4759i	0.122610i	−0.998119	+	0.0613051i	$0.980474\pi$
$168$	0	0
$169$	−156.816	−0.927903
$170$	0	0
$171$	103.285	0.604003
$172$	0	0
$173$	− 10.9926i	− 0.0635410i	−0.999495	−	0.0317705i	$-0.989885\pi$
$173$	− 10.9926i	− 0.0635410i	0.999495	−	0.0317705i	$-0.0101146\pi$
$174$	0	0
$175$	− 27.4197i	− 0.156684i
$176$	0	0
$177$	−122.515	−0.692177
$178$	0	0
$179$	−177.062	−0.989173	−0.494587	−	0.869128i	$-0.664681\pi$
$179$	−177.062	−0.989173	−0.494587	+	0.869128i	$0.664681\pi$
$180$	0	0
$181$	194.021i	1.07194i	0.844237	+	0.535969i	$0.180054\pi$
$181$	194.021i	1.07194i	−0.844237	+	0.535969i	$0.819946\pi$
$182$	0	0
$183$	110.124i	0.601768i
$184$	0	0
$185$	−95.9164	−0.518467
$186$	0	0
$187$	−39.0448	−0.208796
$188$	0	0
$189$	28.4954i	0.150769i
$190$	0	0
$191$	326.808i	1.71104i	0.517772	+	0.855519i	$0.326762\pi$
$191$	326.808i	1.71104i	−0.517772	+	0.855519i	$0.673238\pi$
$192$	0	0
$193$	255.564	1.32416	0.662082	−	0.749431i	$-0.269673\pi$
$193$	255.564	1.32416	0.662082	+	0.749431i	$0.269673\pi$
$194$	0	0
$195$	−69.9088	−0.358506
$196$	0	0
$197$	− 266.053i	− 1.35053i	−0.737577	−	0.675263i	$-0.764030\pi$
$197$	− 266.053i	− 1.35053i	0.737577	−	0.675263i	$-0.235970\pi$
$198$	0	0
$199$	266.589i	1.33964i	0.742521	+	0.669822i	$0.233630\pi$
$199$	266.589i	1.33964i	−0.742521	+	0.669822i	$0.766370\pi$
$200$	0	0
$201$	−102.988	−0.512380
$202$	0	0
$203$	45.6834	0.225041
$204$	0	0
$205$	13.9233i	0.0679187i
$206$	0	0
$207$	92.1767i	0.445298i
$208$	0	0
$209$	−55.9680	−0.267790
$210$	0	0
$211$	50.6253	0.239930	0.119965	−	0.992778i	$-0.461722\pi$
$211$	50.6253	0.239930	0.119965	+	0.992778i	$0.461722\pi$
$212$	0	0
$213$	− 83.9402i	− 0.394085i
$214$	0	0
$215$	− 128.066i	− 0.595654i
$216$	0	0
$217$	30.0790	0.138613
$218$	0	0
$219$	161.319	0.736616
$220$	0	0
$221$	− 433.534i	− 1.96169i
$222$	0	0
$223$	− 357.193i	− 1.60176i	−0.598825	−	0.800880i	$-0.704365\pi$
$223$	− 357.193i	− 1.60176i	0.598825	−	0.800880i	$-0.295635\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	−135.900	−0.598680	−0.299340	−	0.954146i	$-0.596767\pi$
$227$	−135.900	−0.598680	−0.299340	+	0.954146i	$0.596767\pi$
$228$	0	0
$229$	396.479i	1.73135i	0.500608	+	0.865674i	$0.333110\pi$
$229$	396.479i	1.73135i	−0.500608	+	0.865674i	$0.666890\pi$
$230$	0	0
$231$	− 15.4411i	− 0.0668448i
$232$	0	0
$233$	−95.1970	−0.408571	−0.204285	−	0.978911i	$-0.565487\pi$
$233$	−95.1970	−0.408571	−0.204285	+	0.978911i	$0.565487\pi$
$234$	0	0
$235$	−92.9142	−0.395379
$236$	0	0
$237$	104.343i	0.440266i
$238$	0	0
$239$	96.9914i	0.405822i	0.979197	+	0.202911i	$0.0650402\pi$
$239$	96.9914i	0.405822i	−0.979197	+	0.202911i	$0.934960\pi$
$240$	0	0
$241$	36.2570	0.150444	0.0752220	−	0.997167i	$-0.476033\pi$
$241$	36.2570	0.150444	0.0752220	+	0.997167i	$0.476033\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	− 42.3206i	− 0.172737i
$246$	0	0
$247$	− 621.442i	− 2.51596i
$248$	0	0
$249$	−9.85907	−0.0395947
$250$	0	0
$251$	−323.513	−1.28890	−0.644449	−	0.764647i	$-0.722913\pi$
$251$	−323.513	−1.28890	−0.644449	+	0.764647i	$0.722913\pi$
$252$	0	0
$253$	− 49.9488i	− 0.197426i
$254$	0	0
$255$	− 93.0214i	− 0.364790i
$256$	0	0
$257$	403.216	1.56893	0.784467	−	0.620171i	$-0.212937\pi$
$257$	403.216	1.56893	0.784467	+	0.620171i	$0.212937\pi$
$258$	0	0
$259$	235.235	0.908242
$260$	0	0
$261$	− 24.9912i	− 0.0957516i
$262$	0	0
$263$	16.0314i	0.0609561i	0.999535	+	0.0304780i	$0.00970296\pi$
$263$	16.0314i	0.0609561i	−0.999535	+	0.0304780i	$0.990297\pi$
$264$	0	0
$265$	85.3441	0.322053
$266$	0	0
$267$	85.0516	0.318545
$268$	0	0
$269$	261.953i	0.973804i	0.873457	+	0.486902i	$0.161873\pi$
$269$	261.953i	0.973804i	−0.873457	+	0.486902i	$0.838127\pi$
$270$	0	0
$271$	− 25.2639i	− 0.0932246i	−0.998913	−	0.0466123i	$-0.985157\pi$
$271$	− 25.2639i	− 0.0932246i	0.998913	−	0.0466123i	$-0.0148425\pi$
$272$	0	0
$273$	171.451	0.628025
$274$	0	0
$275$	8.12822	0.0295572
$276$	0	0
$277$	− 2.48104i	− 0.00895683i	−0.999990	−	0.00447842i	$-0.998574\pi$
$277$	− 2.48104i	− 0.00895683i	0.999990	−	0.00447842i	$-0.00142553\pi$
$278$	0	0
$279$	− 16.4547i	− 0.0589776i
$280$	0	0
$281$	386.378	1.37501	0.687505	−	0.726180i	$-0.258706\pi$
$281$	386.378	1.37501	0.687505	+	0.726180i	$0.258706\pi$
$282$	0	0
$283$	−115.917	−0.409600	−0.204800	−	0.978804i	$-0.565654\pi$
$283$	−115.917	−0.409600	−0.204800	+	0.978804i	$0.565654\pi$
$284$	0	0
$285$	− 133.340i	− 0.467859i
$286$	0	0
$287$	− 34.1469i	− 0.118979i
$288$	0	0
$289$	287.866	0.996076
$290$	0	0
$291$	62.2203	0.213816
$292$	0	0
$293$	536.585i	1.83135i	0.401922	+	0.915674i	$0.368342\pi$
$293$	536.585i	1.83135i	−0.401922	+	0.915674i	$0.631658\pi$
$294$	0	0
$295$	158.167i	0.536158i
$296$	0	0
$297$	−8.44710	−0.0284414
$298$	0	0
$299$	554.607	1.85487
$300$	0	0
$301$	314.080i	1.04346i
$302$	0	0
$303$	116.619i	0.384882i
$304$	0	0
$305$	142.169	0.466128
$306$	0	0
$307$	332.415	1.08279	0.541393	−	0.840770i	$-0.317897\pi$
$307$	332.415	1.08279	0.541393	+	0.840770i	$0.317897\pi$
$308$	0	0
$309$	127.023i	0.411078i
$310$	0	0
$311$	− 523.655i	− 1.68378i	−0.539651	−	0.841889i	$-0.681444\pi$
$311$	− 523.655i	− 1.68378i	0.539651	−	0.841889i	$-0.318556\pi$
$312$	0	0
$313$	−188.357	−0.601781	−0.300890	−	0.953659i	$-0.597284\pi$
$313$	−188.357	−0.601781	−0.300890	+	0.953659i	$0.597284\pi$
$314$	0	0
$315$	36.7874	0.116785
$316$	0	0
$317$	− 393.769i	− 1.24217i	−0.783742	−	0.621087i	$-0.786692\pi$
$317$	− 393.769i	− 1.24217i	0.783742	−	0.621087i	$-0.213308\pi$
$318$	0	0
$319$	13.5422i	0.0424522i
$320$	0	0
$321$	−267.072	−0.832000
$322$	0	0
$323$	826.897	2.56005
$324$	0	0
$325$	90.2518i	0.277698i
$326$	0	0
$327$	− 209.150i	− 0.639603i
$328$	0	0
$329$	227.872	0.692619
$330$	0	0
$331$	94.6057	0.285818	0.142909	−	0.989736i	$-0.454354\pi$
$331$	94.6057	0.285818	0.142909	+	0.989736i	$0.454354\pi$
$332$	0	0
$333$	− 128.685i	− 0.386443i
$334$	0	0
$335$	132.957i	0.396888i
$336$	0	0
$337$	−520.784	−1.54535	−0.772676	−	0.634800i	$-0.781082\pi$
$337$	−520.784	−1.54535	−0.772676	+	0.634800i	$0.781082\pi$
$338$	0	0
$339$	−362.711	−1.06994
$340$	0	0
$341$	8.91652i	0.0261481i
$342$	0	0
$343$	372.504i	1.08602i
$344$	0	0
$345$	119.000	0.344926
$346$	0	0
$347$	21.6254	0.0623210	0.0311605	−	0.999514i	$-0.490080\pi$
$347$	21.6254	0.0623210	0.0311605	+	0.999514i	$0.490080\pi$
$348$	0	0
$349$	− 241.608i	− 0.692286i	−0.938182	−	0.346143i	$-0.887491\pi$
$349$	− 241.608i	− 0.692286i	0.938182	−	0.346143i	$-0.112509\pi$
$350$	0	0
$351$	− 93.7924i	− 0.267215i
$352$	0	0
$353$	−100.887	−0.285799	−0.142899	−	0.989737i	$-0.545643\pi$
$353$	−100.887	−0.285799	−0.142899	+	0.989737i	$0.545643\pi$
$354$	0	0
$355$	−108.366	−0.305257
$356$	0	0
$357$	228.135i	0.639033i
$358$	0	0
$359$	161.368i	0.449494i	0.974417	+	0.224747i	$0.0721556\pi$
$359$	161.368i	0.449494i	−0.974417	+	0.224747i	$0.927844\pi$
$360$	0	0
$361$	824.301	2.28338
$362$	0	0
$363$	−205.001	−0.564741
$364$	0	0
$365$	− 208.262i	− 0.570580i
$366$	0	0
$367$	− 301.732i	− 0.822157i	−0.911600	−	0.411079i	$-0.865152\pi$
$367$	− 301.732i	− 0.822157i	0.911600	−	0.411079i	$-0.134848\pi$
$368$	0	0
$369$	−18.6801	−0.0506236
$370$	0	0
$371$	−209.306	−0.564167
$372$	0	0
$373$	− 263.001i	− 0.705097i	−0.935794	−	0.352548i	$-0.885315\pi$
$373$	− 263.001i	− 0.705097i	0.935794	−	0.352548i	$-0.114685\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−150.367	−0.398850
$378$	0	0
$379$	9.55298	0.0252057	0.0126029	−	0.999921i	$-0.495988\pi$
$379$	9.55298	0.0252057	0.0126029	+	0.999921i	$0.495988\pi$
$380$	0	0
$381$	− 238.440i	− 0.625826i
$382$	0	0
$383$	− 319.027i	− 0.832968i	−0.909143	−	0.416484i	$-0.863262\pi$
$383$	− 319.027i	− 0.832968i	0.909143	−	0.416484i	$-0.136738\pi$
$384$	0	0
$385$	−19.9344	−0.0517777
$386$	0	0
$387$	171.818	0.443974
$388$	0	0
$389$	− 108.219i	− 0.278198i	−0.990278	−	0.139099i	$-0.955579\pi$
$389$	− 108.219i	− 0.278198i	0.990278	−	0.139099i	$-0.0444206\pi$
$390$	0	0
$391$	737.967i	1.88738i
$392$	0	0
$393$	−52.4845	−0.133548
$394$	0	0
$395$	134.706	0.341028
$396$	0	0
$397$	− 533.165i	− 1.34299i	−0.741011	−	0.671493i	$-0.765654\pi$
$397$	− 533.165i	− 1.34299i	0.741011	−	0.671493i	$-0.234346\pi$
$398$	0	0
$399$	327.015i	0.819587i
$400$	0	0
$401$	610.423	1.52225	0.761126	−	0.648604i	$-0.224647\pi$
$401$	610.423	1.52225	0.761126	+	0.648604i	$0.224647\pi$
$402$	0	0
$403$	−99.0047	−0.245669
$404$	0	0
$405$	− 20.1246i	− 0.0496904i
$406$	0	0
$407$	69.7322i	0.171332i
$408$	0	0
$409$	−174.487	−0.426618	−0.213309	−	0.976985i	$-0.568424\pi$
$409$	−174.487	−0.426618	−0.213309	+	0.976985i	$0.568424\pi$
$410$	0	0
$411$	−322.277	−0.784129
$412$	0	0
$413$	− 387.903i	− 0.939232i
$414$	0	0
$415$	12.7280i	0.0306699i
$416$	0	0
$417$	−10.2583	−0.0246002
$418$	0	0
$419$	−35.8925	−0.0856622	−0.0428311	−	0.999082i	$-0.513638\pi$
$419$	−35.8925	−0.0856622	−0.0428311	+	0.999082i	$0.513638\pi$
$420$	0	0
$421$	236.757i	0.562368i	0.959654	+	0.281184i	$0.0907271\pi$
$421$	236.757i	0.562368i	−0.959654	+	0.281184i	$0.909273\pi$
$422$	0	0
$423$	− 124.657i	− 0.294698i
$424$	0	0
$425$	−120.090	−0.282565
$426$	0	0
$427$	−348.669	−0.816554
$428$	0	0
$429$	50.8244i	0.118472i
$430$	0	0
$431$	195.847i	0.454403i	0.973848	+	0.227201i	$0.0729575\pi$
$431$	195.847i	0.454403i	−0.973848	+	0.227201i	$0.927042\pi$
$432$	0	0
$433$	−66.2067	−0.152902	−0.0764512	−	0.997073i	$-0.524359\pi$
$433$	−66.2067	−0.152902	−0.0764512	+	0.997073i	$0.524359\pi$
$434$	0	0
$435$	−32.2634	−0.0741688
$436$	0	0
$437$	1057.83i	2.42065i
$438$	0	0
$439$	− 551.797i	− 1.25694i	−0.777834	−	0.628470i	$-0.783681\pi$
$439$	− 551.797i	− 1.25694i	0.777834	−	0.628470i	$-0.216319\pi$
$440$	0	0
$441$	56.7790	0.128751
$442$	0	0
$443$	277.959	0.627446	0.313723	−	0.949515i	$-0.398424\pi$
$443$	277.959	0.627446	0.313723	+	0.949515i	$0.398424\pi$
$444$	0	0
$445$	− 109.801i	− 0.246744i
$446$	0	0
$447$	408.567i	0.914019i
$448$	0	0
$449$	−147.609	−0.328752	−0.164376	−	0.986398i	$-0.552561\pi$
$449$	−147.609	−0.328752	−0.164376	+	0.986398i	$0.552561\pi$
$450$	0	0
$451$	10.1224	0.0224444
$452$	0	0
$453$	98.6198i	0.217704i
$454$	0	0
$455$	− 221.342i	− 0.486466i
$456$	0	0
$457$	−485.661	−1.06272	−0.531358	−	0.847147i	$-0.678318\pi$
$457$	−485.661	−1.06272	−0.531358	+	0.847147i	$0.678318\pi$
$458$	0	0
$459$	124.801	0.271898
$460$	0	0
$461$	− 447.963i	− 0.971721i	−0.874036	−	0.485861i	$-0.838506\pi$
$461$	− 447.963i	− 0.971721i	0.874036	−	0.485861i	$-0.161494\pi$
$462$	0	0
$463$	762.712i	1.64733i	0.567080	+	0.823663i	$0.308073\pi$
$463$	762.712i	1.64733i	−0.567080	+	0.823663i	$0.691927\pi$
$464$	0	0
$465$	−21.2430	−0.0456838
$466$	0	0
$467$	838.223	1.79491	0.897455	−	0.441106i	$-0.145414\pi$
$467$	838.223	1.79491	0.897455	+	0.441106i	$0.145414\pi$
$468$	0	0
$469$	− 326.077i	− 0.695261i
$470$	0	0
$471$	− 526.825i	− 1.11852i
$472$	0	0
$473$	−93.1050	−0.196839
$474$	0	0
$475$	−172.141	−0.362402
$476$	0	0
$477$	114.501i	0.240044i
$478$	0	0
$479$	150.795i	0.314812i	0.987534	+	0.157406i	$0.0503131\pi$
$479$	150.795i	0.314812i	−0.987534	+	0.157406i	$0.949687\pi$
$480$	0	0
$481$	−774.273	−1.60971
$482$	0	0
$483$	−291.846	−0.604236
$484$	0	0
$485$	− 80.3261i	− 0.165621i
$486$	0	0
$487$	148.464i	0.304855i	0.988315	+	0.152427i	$0.0487090\pi$
$487$	148.464i	0.304855i	−0.988315	+	0.152427i	$0.951291\pi$
$488$	0	0
$489$	−430.711	−0.880801
$490$	0	0
$491$	−76.8116	−0.156439	−0.0782196	−	0.996936i	$-0.524924\pi$
$491$	−76.8116	−0.156439	−0.0782196	+	0.996936i	$0.524924\pi$
$492$	0	0
$493$	− 200.079i	− 0.405841i
$494$	0	0
$495$	10.9052i	0.0220306i
$496$	0	0
$497$	265.768	0.534744
$498$	0	0
$499$	−564.248	−1.13076	−0.565379	−	0.824832i	$-0.691270\pi$
$499$	−564.248	−1.13076	−0.565379	+	0.824832i	$0.691270\pi$
$500$	0	0
$501$	35.4653i	0.0707890i
$502$	0	0
$503$	− 223.999i	− 0.445327i	−0.974895	−	0.222663i	$-0.928525\pi$
$503$	− 223.999i	− 0.445327i	0.974895	−	0.222663i	$-0.0714750\pi$
$504$	0	0
$505$	150.555	0.298128
$506$	0	0
$507$	−271.613	−0.535725
$508$	0	0
$509$	174.435i	0.342701i	0.985210	+	0.171350i	$0.0548130\pi$
$509$	174.435i	0.342701i	−0.985210	+	0.171350i	$0.945187\pi$
$510$	0	0
$511$	510.761i	0.999532i
$512$	0	0
$513$	178.894	0.348722
$514$	0	0
$515$	163.986	0.318420
$516$	0	0
$517$	67.5496i	0.130657i
$518$	0	0
$519$	− 19.0397i	− 0.0366854i
$520$	0	0
$521$	478.465	0.918358	0.459179	−	0.888344i	$-0.348144\pi$
$521$	478.465	0.918358	0.459179	+	0.888344i	$0.348144\pi$
$522$	0	0
$523$	491.609	0.939979	0.469990	−	0.882672i	$-0.344258\pi$
$523$	491.609	0.939979	0.469990	+	0.882672i	$0.344258\pi$
$524$	0	0
$525$	− 47.4924i	− 0.0904616i
$526$	0	0
$527$	− 131.737i	− 0.249975i
$528$	0	0
$529$	−415.059	−0.784611
$530$	0	0
$531$	−212.203	−0.399628
$532$	0	0
$533$	112.394i	0.210871i
$534$	0	0
$535$	344.788i	0.644464i
$536$	0	0
$537$	−306.680	−0.571099
$538$	0	0
$539$	−30.7675	−0.0570826
$540$	0	0
$541$	166.802i	0.308322i	0.988046	+	0.154161i	$0.0492674\pi$
$541$	166.802i	0.308322i	−0.988046	+	0.154161i	$0.950733\pi$
$542$	0	0
$543$	336.054i	0.618884i
$544$	0	0
$545$	−270.012	−0.495434
$546$	0	0
$547$	−166.395	−0.304195	−0.152098	−	0.988365i	$-0.548603\pi$
$547$	−166.395	−0.304195	−0.152098	+	0.988365i	$0.548603\pi$
$548$	0	0
$549$	190.740i	0.347431i
$550$	0	0
$551$	− 286.800i	− 0.520508i
$552$	0	0
$553$	−330.366	−0.597407
$554$	0	0
$555$	−166.132	−0.299337
$556$	0	0
$557$	267.280i	0.479856i	0.970791	+	0.239928i	$0.0771238\pi$
$557$	267.280i	0.479856i	−0.970791	+	0.239928i	$0.922876\pi$
$558$	0	0
$559$	− 1033.79i	− 1.84936i
$560$	0	0
$561$	−67.6275	−0.120548
$562$	0	0
$563$	−637.679	−1.13264	−0.566322	−	0.824184i	$-0.691634\pi$
$563$	−637.679	−1.13264	−0.566322	+	0.824184i	$0.691634\pi$
$564$	0	0
$565$	468.258i	0.828776i
$566$	0	0
$567$	49.3555i	0.0870468i
$568$	0	0
$569$	58.7735	0.103293	0.0516463	−	0.998665i	$-0.483553\pi$
$569$	58.7735	0.103293	0.0516463	+	0.998665i	$0.483553\pi$
$570$	0	0
$571$	3.25216	0.00569555	0.00284778	−	0.999996i	$-0.499094\pi$
$571$	3.25216	0.00569555	0.00284778	+	0.999996i	$0.499094\pi$
$572$	0	0
$573$	566.048i	0.987868i
$574$	0	0
$575$	− 153.628i	− 0.267179i
$576$	0	0
$577$	366.712	0.635549	0.317775	−	0.948166i	$-0.397064\pi$
$577$	366.712	0.635549	0.317775	+	0.948166i	$0.397064\pi$
$578$	0	0
$579$	442.649	0.764506
$580$	0	0
$581$	− 31.2154i	− 0.0537270i
$582$	0	0
$583$	− 62.0460i	− 0.106425i
$584$	0	0
$585$	−121.086	−0.206984
$586$	0	0
$587$	−203.754	−0.347111	−0.173556	−	0.984824i	$-0.555526\pi$
$587$	−203.754	−0.347111	−0.173556	+	0.984824i	$0.555526\pi$
$588$	0	0
$589$	− 188.836i	− 0.320604i
$590$	0	0
$591$	− 460.818i	− 0.779726i
$592$	0	0
$593$	291.378	0.491362	0.245681	−	0.969351i	$-0.420988\pi$
$593$	291.378	0.491362	0.245681	+	0.969351i	$0.420988\pi$
$594$	0	0
$595$	294.521	0.494992
$596$	0	0
$597$	461.746i	0.773444i
$598$	0	0
$599$	− 855.966i	− 1.42899i	−0.699640	−	0.714495i	$-0.746656\pi$
$599$	− 855.966i	− 1.42899i	0.699640	−	0.714495i	$-0.253344\pi$
$600$	0	0
$601$	−426.264	−0.709258	−0.354629	−	0.935007i	$-0.615393\pi$
$601$	−426.264	−0.709258	−0.354629	+	0.935007i	$0.615393\pi$
$602$	0	0
$603$	−178.381	−0.295823
$604$	0	0
$605$	264.655i	0.437446i
$606$	0	0
$607$	− 410.035i	− 0.675510i	−0.941234	−	0.337755i	$-0.890332\pi$
$607$	− 410.035i	− 0.675510i	0.941234	−	0.337755i	$-0.109668\pi$
$608$	0	0
$609$	79.1259	0.129928
$610$	0	0
$611$	−750.037	−1.22756
$612$	0	0
$613$	− 252.628i	− 0.412118i	−0.978540	−	0.206059i	$-0.933936\pi$
$613$	− 252.628i	− 0.412118i	0.978540	−	0.206059i	$-0.0660639\pi$
$614$	0	0
$615$	24.1159i	0.0392129i
$616$	0	0
$617$	234.903	0.380718	0.190359	−	0.981715i	$-0.439035\pi$
$617$	234.903	0.380718	0.190359	+	0.981715i	$0.439035\pi$
$618$	0	0
$619$	−412.764	−0.666824	−0.333412	−	0.942781i	$-0.608200\pi$
$619$	−412.764	−0.666824	−0.333412	+	0.942781i	$0.608200\pi$
$620$	0	0
$621$	159.655i	0.257093i
$622$	0	0
$623$	269.287i	0.432242i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	−96.9394	−0.154608
$628$	0	0
$629$	− 1030.26i	− 1.63793i
$630$	0	0
$631$	969.308i	1.53615i	0.640363	+	0.768073i	$0.278784\pi$
$631$	969.308i	1.53615i	−0.640363	+	0.768073i	$0.721216\pi$
$632$	0	0
$633$	87.6856	0.138524
$634$	0	0
$635$	−307.824	−0.484763
$636$	0	0
$637$	− 341.627i	− 0.536307i
$638$	0	0
$639$	− 145.389i	− 0.227525i
$640$	0	0
$641$	−805.292	−1.25631	−0.628153	−	0.778090i	$-0.716189\pi$
$641$	−805.292	−1.25631	−0.628153	+	0.778090i	$0.716189\pi$
$642$	0	0
$643$	−887.377	−1.38006	−0.690028	−	0.723782i	$-0.742402\pi$
$643$	−887.377	−1.38006	−0.690028	+	0.723782i	$0.742402\pi$
$644$	0	0
$645$	− 221.816i	− 0.343901i
$646$	0	0
$647$	1129.74i	1.74612i	0.487614	+	0.873059i	$0.337867\pi$
$647$	1129.74i	1.74612i	−0.487614	+	0.873059i	$0.662133\pi$
$648$	0	0
$649$	114.989	0.177178
$650$	0	0
$651$	52.0983	0.0800281
$652$	0	0
$653$	384.844i	0.589348i	0.955598	+	0.294674i	$0.0952111\pi$
$653$	384.844i	0.589348i	−0.955598	+	0.294674i	$0.904789\pi$
$654$	0	0
$655$	67.7572i	0.103446i
$656$	0	0
$657$	279.412	0.425285
$658$	0	0
$659$	−38.8992	−0.0590276	−0.0295138	−	0.999564i	$-0.509396\pi$
$659$	−38.8992	−0.0590276	−0.0295138	+	0.999564i	$0.509396\pi$
$660$	0	0
$661$	877.861i	1.32808i	0.747697	+	0.664040i	$0.231159\pi$
$661$	877.861i	1.32808i	−0.747697	+	0.664040i	$0.768841\pi$
$662$	0	0
$663$	− 750.903i	− 1.13258i
$664$	0	0
$665$	422.175	0.634850
$666$	0	0
$667$	255.956	0.383742
$668$	0	0
$669$	− 618.676i	− 0.924777i
$670$	0	0
$671$	− 103.358i	− 0.154036i
$672$	0	0
$673$	−343.747	−0.510768	−0.255384	−	0.966840i	$-0.582202\pi$
$673$	−343.747	−0.510768	−0.255384	+	0.966840i	$0.582202\pi$
$674$	0	0
$675$	−25.9808	−0.0384900
$676$	0	0
$677$	404.932i	0.598126i	0.954233	+	0.299063i	$0.0966741\pi$
$677$	404.932i	0.598126i	−0.954233	+	0.299063i	$0.903326\pi$
$678$	0	0
$679$	196.999i	0.290132i
$680$	0	0
$681$	−235.386	−0.345648
$682$	0	0
$683$	−145.169	−0.212546	−0.106273	−	0.994337i	$-0.533892\pi$
$683$	−145.169	−0.212546	−0.106273	+	0.994337i	$0.533892\pi$
$684$	0	0
$685$	416.058i	0.607384i
$686$	0	0
$687$	686.721i	0.999594i
$688$	0	0
$689$	688.929	0.999896
$690$	0	0
$691$	104.860	0.151751	0.0758757	−	0.997117i	$-0.475825\pi$
$691$	104.860	0.151751	0.0758757	+	0.997117i	$0.475825\pi$
$692$	0	0
$693$	− 26.7448i	− 0.0385928i
$694$	0	0
$695$	13.2434i	0.0190552i
$696$	0	0
$697$	−149.553	−0.214567
$698$	0	0
$699$	−164.886	−0.235888
$700$	0	0
$701$	− 88.5728i	− 0.126352i	−0.998002	−	0.0631760i	$-0.979877\pi$
$701$	− 88.5728i	− 0.126352i	0.998002	−	0.0631760i	$-0.0201229\pi$
$702$	0	0
$703$	− 1476.80i	− 2.10071i
$704$	0	0
$705$	−160.932	−0.228272
$706$	0	0
$707$	−369.235	−0.522256
$708$	0	0
$709$	295.500i	0.416784i	0.978045	+	0.208392i	$0.0668230\pi$
$709$	295.500i	0.416784i	−0.978045	+	0.208392i	$0.933177\pi$
$710$	0	0
$711$	180.727i	0.254187i
$712$	0	0
$713$	168.527	0.236363
$714$	0	0
$715$	65.6140	0.0917678
$716$	0	0
$717$	167.994i	0.234301i
$718$	0	0
$719$	464.090i	0.645467i	0.946490	+	0.322733i	$0.104602\pi$
$719$	464.090i	0.645467i	−0.946490	+	0.322733i	$0.895398\pi$
$720$	0	0
$721$	−402.175	−0.557802
$722$	0	0
$723$	62.7989	0.0868588
$724$	0	0
$725$	41.6519i	0.0574509i
$726$	0	0
$727$	1052.48i	1.44771i	0.689952	+	0.723855i	$0.257631\pi$
$727$	1052.48i	1.44771i	−0.689952	+	0.723855i	$0.742369\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	1375.58	1.88177
$732$	0	0
$733$	− 82.5148i	− 0.112571i	−0.998415	−	0.0562857i	$-0.982074\pi$
$733$	− 82.5148i	− 0.112571i	0.998415	−	0.0562857i	$-0.0179258\pi$
$734$	0	0
$735$	− 73.3014i	− 0.0997298i
$736$	0	0
$737$	96.6614	0.131155
$738$	0	0
$739$	−214.733	−0.290572	−0.145286	−	0.989390i	$-0.546410\pi$
$739$	−214.733	−0.290572	−0.145286	+	0.989390i	$0.546410\pi$
$740$	0	0
$741$	− 1076.37i	− 1.45259i
$742$	0	0
$743$	− 96.3747i	− 0.129710i	−0.997895	−	0.0648551i	$-0.979341\pi$
$743$	− 96.3747i	− 0.129710i	0.997895	−	0.0648551i	$-0.0206585\pi$
$744$	0	0
$745$	527.457	0.707996
$746$	0	0
$747$	−17.0764	−0.0228600
$748$	0	0
$749$	− 845.592i	− 1.12896i
$750$	0	0
$751$	− 547.767i	− 0.729384i	−0.931128	−	0.364692i	$-0.881174\pi$
$751$	− 547.767i	− 0.729384i	0.931128	−	0.364692i	$-0.118826\pi$
$752$	0	0
$753$	−560.342	−0.744146
$754$	0	0
$755$	127.318	0.168633
$756$	0	0
$757$	− 516.109i	− 0.681782i	−0.940103	−	0.340891i	$-0.889271\pi$
$757$	− 516.109i	− 0.681782i	0.940103	−	0.340891i	$-0.110729\pi$
$758$	0	0
$759$	− 86.5139i	− 0.113984i
$760$	0	0
$761$	−446.601	−0.586861	−0.293430	−	0.955980i	$-0.594797\pi$
$761$	−446.601	−0.586861	−0.293430	+	0.955980i	$0.594797\pi$
$762$	0	0
$763$	662.202	0.867893
$764$	0	0
$765$	− 161.118i	− 0.210612i
$766$	0	0
$767$	1276.78i	1.66464i
$768$	0	0
$769$	−253.980	−0.330273	−0.165136	−	0.986271i	$-0.552806\pi$
$769$	−253.980	−0.330273	−0.165136	+	0.986271i	$0.552806\pi$
$770$	0	0
$771$	698.391	0.905824
$772$	0	0
$773$	226.748i	0.293335i	0.989186	+	0.146667i	$0.0468547\pi$
$773$	226.748i	0.293335i	−0.989186	+	0.146667i	$0.953145\pi$
$774$	0	0
$775$	27.4246i	0.0353865i
$776$	0	0
$777$	407.438	0.524374
$778$	0	0
$779$	−214.374	−0.275192
$780$	0	0
$781$	78.7835i	0.100875i
$782$	0	0
$783$	− 43.2860i	− 0.0552822i
$784$	0	0
$785$	−680.128	−0.866405
$786$	0	0
$787$	86.1883	0.109515	0.0547575	−	0.998500i	$-0.482561\pi$
$787$	86.1883	0.109515	0.0547575	+	0.998500i	$0.482561\pi$
$788$	0	0
$789$	27.7673i	0.0351930i
$790$	0	0
$791$	− 1148.40i	− 1.45183i
$792$	0	0
$793$	1147.64	1.44721
$794$	0	0
$795$	147.820	0.185937
$796$	0	0
$797$	661.678i	0.830211i	0.909773	+	0.415105i	$0.136255\pi$
$797$	661.678i	0.830211i	−0.909773	+	0.415105i	$0.863745\pi$
$798$	0	0
$799$	− 998.009i	− 1.24907i
$800$	0	0
$801$	147.314	0.183912
$802$	0	0
$803$	−151.408	−0.188553
$804$	0	0
$805$	376.771i	0.468039i
$806$	0	0
$807$	453.716i	0.562226i
$808$	0	0
$809$	−1095.50	−1.35414	−0.677070	−	0.735919i	$-0.736750\pi$
$809$	−1095.50	−1.35414	−0.677070	+	0.735919i	$0.736750\pi$
$810$	0	0
$811$	743.013	0.916169	0.458084	−	0.888909i	$-0.348536\pi$
$811$	743.013	0.916169	0.458084	+	0.888909i	$0.348536\pi$
$812$	0	0
$813$	− 43.7583i	− 0.0538232i
$814$	0	0
$815$	556.046i	0.682265i
$816$	0	0
$817$	1971.79	2.41346
$818$	0	0
$819$	296.962	0.362591
$820$	0	0
$821$	− 1269.96i	− 1.54685i	−0.633888	−	0.773425i	$-0.718542\pi$
$821$	− 1269.96i	− 1.54685i	0.633888	−	0.773425i	$-0.281458\pi$
$822$	0	0
$823$	− 297.950i	− 0.362029i	−0.983480	−	0.181015i	$-0.942062\pi$
$823$	− 297.950i	− 0.362029i	0.983480	−	0.181015i	$-0.0579381\pi$
$824$	0	0
$825$	14.0785	0.0170648
$826$	0	0
$827$	386.605	0.467478	0.233739	−	0.972299i	$-0.424904\pi$
$827$	386.605	0.467478	0.233739	+	0.972299i	$0.424904\pi$
$828$	0	0
$829$	− 606.395i	− 0.731478i	−0.930717	−	0.365739i	$-0.880816\pi$
$829$	− 606.395i	− 0.731478i	0.930717	−	0.365739i	$-0.119184\pi$
$830$	0	0
$831$	− 4.29729i	− 0.00517123i
$832$	0	0
$833$	454.573	0.545706
$834$	0	0
$835$	45.7855	0.0548329
$836$	0	0
$837$	− 28.5004i	− 0.0340507i
$838$	0	0
$839$	634.313i	0.756035i	0.925799	+	0.378017i	$0.123394\pi$
$839$	634.313i	0.756035i	−0.925799	+	0.378017i	$0.876606\pi$
$840$	0	0
$841$	771.605	0.917485
$842$	0	0
$843$	669.226	0.793862
$844$	0	0
$845$	350.651i	0.414971i
$846$	0	0
$847$	− 649.065i	− 0.766310i
$848$	0	0
$849$	−200.774	−0.236482
$850$	0	0
$851$	1317.98	1.54874
$852$	0	0
$853$	− 350.351i	− 0.410728i	−0.978686	−	0.205364i	$-0.934162\pi$
$853$	− 350.351i	− 0.410728i	0.978686	−	0.205364i	$-0.0658379\pi$
$854$	0	0
$855$	− 230.951i	− 0.270119i
$856$	0	0
$857$	−582.739	−0.679976	−0.339988	−	0.940430i	$-0.610423\pi$
$857$	−582.739	−0.679976	−0.339988	+	0.940430i	$0.610423\pi$
$858$	0	0
$859$	91.0452	0.105990	0.0529948	−	0.998595i	$-0.483123\pi$
$859$	91.0452	0.105990	0.0529948	+	0.998595i	$0.483123\pi$
$860$	0	0
$861$	− 59.1442i	− 0.0686924i
$862$	0	0
$863$	311.660i	0.361135i	0.983563	+	0.180568i	$0.0577935\pi$
$863$	311.660i	0.361135i	−0.983563	+	0.180568i	$0.942207\pi$
$864$	0	0
$865$	−24.5802	−0.0284164
$866$	0	0
$867$	498.598	0.575084
$868$	0	0
$869$	− 97.9328i	− 0.112696i
$870$	0	0
$871$	1073.28i	1.23224i
$872$	0	0
$873$	107.769	0.123446
$874$	0	0
$875$	−61.3124	−0.0700713
$876$	0	0
$877$	1264.84i	1.44223i	0.692815	+	0.721115i	$0.256370\pi$
$877$	1264.84i	1.44223i	−0.692815	+	0.721115i	$0.743630\pi$
$878$	0	0
$879$	929.392i	1.05733i
$880$	0	0
$881$	−1185.61	−1.34576	−0.672880	−	0.739752i	$-0.734943\pi$
$881$	−1185.61	−1.34576	−0.672880	+	0.739752i	$0.734943\pi$
$882$	0	0
$883$	1734.26	1.96405	0.982026	−	0.188745i	$-0.0604420\pi$
$883$	1734.26	1.96405	0.982026	+	0.188745i	$0.0604420\pi$
$884$	0	0
$885$	273.953i	0.309551i
$886$	0	0
$887$	− 780.604i	− 0.880050i	−0.897986	−	0.440025i	$-0.854970\pi$
$887$	− 780.604i	− 0.880050i	0.897986	−	0.440025i	$-0.145030\pi$
$888$	0	0
$889$	754.937	0.849198
$890$	0	0
$891$	−14.6308	−0.0164207
$892$	0	0
$893$	− 1430.58i	− 1.60199i
$894$	0	0
$895$	395.923i	0.442372i
$896$	0	0
$897$	960.608	1.07091
$898$	0	0
$899$	−45.6914	−0.0508247
$900$	0	0
$901$	916.697i	1.01742i
$902$	0	0
$903$	544.003i	0.602439i
$904$	0	0
$905$	433.844	0.479386
$906$	0	0
$907$	−1546.71	−1.70531	−0.852654	−	0.522476i	$-0.825008\pi$
$907$	−1546.71	−1.70531	−0.852654	+	0.522476i	$0.825008\pi$
$908$	0	0
$909$	201.990i	0.222212i
$910$	0	0
$911$	− 739.818i	− 0.812094i	−0.913852	−	0.406047i	$-0.866907\pi$
$911$	− 739.818i	− 0.812094i	0.913852	−	0.406047i	$-0.133093\pi$
$912$	0	0
$913$	9.25339	0.0101352
$914$	0	0
$915$	246.244	0.269119
$916$	0	0
$917$	− 166.174i	− 0.181215i
$918$	0	0
$919$	− 334.138i	− 0.363589i	−0.983337	−	0.181795i	$-0.941809\pi$
$919$	− 334.138i	− 0.363589i	0.983337	−	0.181795i	$-0.0581906\pi$
$920$	0	0
$921$	575.760	0.625146
$922$	0	0
$923$	−874.773	−0.947750
$924$	0	0
$925$	214.476i	0.231866i
$926$	0	0
$927$	220.010i	0.237336i
$928$	0	0
$929$	668.207	0.719276	0.359638	−	0.933092i	$-0.382900\pi$
$929$	668.207	0.719276	0.359638	+	0.933092i	$0.382900\pi$
$930$	0	0
$931$	651.600	0.699892
$932$	0	0
$933$	− 906.997i	− 0.972129i
$934$	0	0
$935$	87.3068i	0.0933762i
$936$	0	0
$937$	−1204.35	−1.28532	−0.642661	−	0.766151i	$-0.722170\pi$
$937$	−1204.35	−1.28532	−0.642661	+	0.766151i	$0.722170\pi$
$938$	0	0
$939$	−326.245	−0.347438
$940$	0	0
$941$	794.818i	0.844653i	0.906444	+	0.422326i	$0.138786\pi$
$941$	794.818i	0.844653i	−0.906444	+	0.422326i	$0.861214\pi$
$942$	0	0
$943$	− 191.319i	− 0.202883i
$944$	0	0
$945$	63.7177	0.0674261
$946$	0	0
$947$	−505.479	−0.533769	−0.266884	−	0.963729i	$-0.585994\pi$
$947$	−505.479	−0.533769	−0.266884	+	0.963729i	$0.585994\pi$
$948$	0	0
$949$	− 1681.17i	− 1.77151i
$950$	0	0
$951$	− 682.028i	− 0.717169i
$952$	0	0
$953$	−586.218	−0.615129	−0.307564	−	0.951527i	$-0.599514\pi$
$953$	−586.218	−0.615129	−0.307564	+	0.951527i	$0.599514\pi$
$954$	0	0
$955$	730.765	0.765199
$956$	0	0
$957$	23.4559i	0.0245098i
$958$	0	0
$959$	− 1020.38i	− 1.06400i
$960$	0	0
$961$	930.916	0.968695
$962$	0	0
$963$	−462.582	−0.480355
$964$	0	0
$965$	− 571.458i	− 0.592184i
$966$	0	0
$967$	598.983i	0.619424i	0.950830	+	0.309712i	$0.100233\pi$
$967$	598.983i	0.619424i	−0.950830	+	0.309712i	$0.899767\pi$
$968$	0	0
$969$	1432.23	1.47805
$970$	0	0
$971$	−335.550	−0.345571	−0.172786	−	0.984959i	$-0.555277\pi$
$971$	−335.550	−0.345571	−0.172786	+	0.984959i	$0.555277\pi$
$972$	0	0
$973$	− 32.4793i	− 0.0333806i
$974$	0	0
$975$	156.321i	0.160329i
$976$	0	0
$977$	1009.81	1.03359	0.516794	−	0.856110i	$-0.327125\pi$
$977$	1009.81	1.03359	0.516794	+	0.856110i	$0.327125\pi$
$978$	0	0
$979$	−79.8266	−0.0815389
$980$	0	0
$981$	− 362.259i	− 0.369275i
$982$	0	0
$983$	− 1449.61i	− 1.47468i	−0.675521	−	0.737340i	$-0.736081\pi$
$983$	− 1449.61i	− 1.47468i	0.675521	−	0.737340i	$-0.263919\pi$
$984$	0	0
$985$	−594.914	−0.603973
$986$	0	0
$987$	394.685	0.399884
$988$	0	0
$989$	1759.73i	1.77931i
$990$	0	0
$991$	− 1085.11i	− 1.09497i	−0.836816	−	0.547485i	$-0.815585\pi$
$991$	− 1085.11i	− 1.09497i	0.836816	−	0.547485i	$-0.184415\pi$
$992$	0	0
$993$	163.862	0.165017
$994$	0	0
$995$	596.112	0.599107
$996$	0	0
$997$	− 1471.19i	− 1.47561i	−0.675011	−	0.737807i	$-0.735861\pi$
$997$	− 1471.19i	− 1.47561i	0.675011	−	0.737807i	$-0.264139\pi$
$998$	0	0
$999$	− 222.890i	− 0.223113i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.3.g.a.271.11		16
3.2	odd	2		1440.3.g.c.271.14		16
4.3	odd	2		120.3.g.a.91.10	yes	16
5.2	odd	4		2400.3.p.b.1999.5		32
5.3	odd	4		2400.3.p.b.1999.28		32
5.4	even	2		2400.3.g.b.751.4		16
8.3	odd	2	inner	480.3.g.a.271.14		16
8.5	even	2		120.3.g.a.91.9	✓	16
12.11	even	2		360.3.g.c.91.7		16
20.3	even	4		600.3.p.b.499.32		32
20.7	even	4		600.3.p.b.499.1		32
20.19	odd	2		600.3.g.d.451.7		16
24.5	odd	2		360.3.g.c.91.8		16
24.11	even	2		1440.3.g.c.271.3		16
40.3	even	4		2400.3.p.b.1999.6		32
40.13	odd	4		600.3.p.b.499.2		32
40.19	odd	2		2400.3.g.b.751.5		16
40.27	even	4		2400.3.p.b.1999.27		32
40.29	even	2		600.3.g.d.451.8		16
40.37	odd	4		600.3.p.b.499.31		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.g.a.91.9	✓	16	8.5	even	2
120.3.g.a.91.10	yes	16	4.3	odd	2
360.3.g.c.91.7		16	12.11	even	2
360.3.g.c.91.8		16	24.5	odd	2
480.3.g.a.271.11		16	1.1	even	1	trivial
480.3.g.a.271.14		16	8.3	odd	2	inner
600.3.g.d.451.7		16	20.19	odd	2
600.3.g.d.451.8		16	40.29	even	2
600.3.p.b.499.1		32	20.7	even	4
600.3.p.b.499.2		32	40.13	odd	4
600.3.p.b.499.31		32	40.37	odd	4
600.3.p.b.499.32		32	20.3	even	4
1440.3.g.c.271.3		16	24.11	even	2
1440.3.g.c.271.14		16	3.2	odd	2
2400.3.g.b.751.4		16	5.4	even	2
2400.3.g.b.751.5		16	40.19	odd	2
2400.3.p.b.1999.5		32	5.2	odd	4
2400.3.p.b.1999.6		32	40.3	even	4
2400.3.p.b.1999.27		32	40.27	even	4
2400.3.p.b.1999.28		32	5.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	480.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -2.23607i q^{5} +5.48395i q^{7} +3.00000 q^{9} -1.62564 q^{11} -18.0504i q^{13} -3.87298i q^{15} +24.0180 q^{17} +34.4282 q^{19} +9.49847i q^{21} +30.7256i q^{23} -5.00000 q^{25} +5.19615 q^{27} -8.33039i q^{29} -5.48491i q^{31} -2.81570 q^{33} +12.2625 q^{35} -42.8951i q^{37} -31.2641i q^{39} -6.22670 q^{41} +57.2727 q^{43} -6.70820i q^{45} -41.5525i q^{47} +18.9263 q^{49} +41.6005 q^{51} +38.1670i q^{53} +3.63505i q^{55} +59.6314 q^{57} -70.7342 q^{59} +63.5799i q^{61} +16.4518i q^{63} -40.3618 q^{65} -59.4604 q^{67} +53.2182i q^{69} -48.4629i q^{71} +93.1375 q^{73} -8.66025 q^{75} -8.91495i q^{77} +60.2424i q^{79} +9.00000 q^{81} -5.69214 q^{83} -53.7060i q^{85} -14.4287i q^{87} +49.1046 q^{89} +98.9872 q^{91} -9.50015i q^{93} -76.9838i q^{95} +35.9229 q^{97} -4.87693 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9} - 64 q^{11} + 32 q^{19} - 80 q^{25} + 192 q^{43} - 80 q^{49} - 96 q^{51} + 96 q^{57} - 128 q^{59} + 64 q^{67} - 160 q^{73} + 144 q^{81} + 192 q^{91} - 224 q^{97} - 192 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 64 * q^11 + 32 * q^19 - 80 * q^25 + 192 * q^43 - 80 * q^49 - 96 * q^51 + 96 * q^57 - 128 * q^59 + 64 * q^67 - 160 * q^73 + 144 * q^81 + 192 * q^91 - 224 * q^97 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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