LMFDB - Embedded newform 640.3.b.a.511.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,3,Mod(511,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, 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("640.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	640.b (of order $2$, degree $1$, 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:	$17.4387369191$
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} + 24x^{14} + 228x^{12} + 1110x^{10} + 2970x^{8} + 4308x^{6} + 3085x^{4} + 882x^{2} + 81 $ x^16 + 24x^14 + 228x^12 + 1110x^10 + 2970x^8 + 4308x^6 + 3085x^4 + 882*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{44} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.11
Root			$1.14902i$ of defining polynomial
Character	$\chi$	$=$	640.511
Dual form			640.3.b.a.511.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.29803i q^{3} -2.23607 q^{5} +2.72697i q^{7} +3.71904 q^{9} +O(q^{10})$	$q+2.29803i q^{3} -2.23607 q^{5} +2.72697i q^{7} +3.71904 q^{9} -19.3454i q^{11} +18.8033 q^{13} -5.13856i q^{15} -23.5759 q^{17} -27.8780i q^{19} -6.26667 q^{21} +8.78218i q^{23} +5.00000 q^{25} +29.2288i q^{27} +23.9370 q^{29} -9.69022i q^{31} +44.4563 q^{33} -6.09769i q^{35} +69.6453 q^{37} +43.2106i q^{39} +32.7276 q^{41} -28.2097i q^{43} -8.31602 q^{45} +37.6454i q^{47} +41.5636 q^{49} -54.1783i q^{51} -42.7329 q^{53} +43.2575i q^{55} +64.0645 q^{57} +18.3369i q^{59} +35.2430 q^{61} +10.1417i q^{63} -42.0454 q^{65} -47.0790i q^{67} -20.1818 q^{69} +85.2253i q^{71} +15.1258 q^{73} +11.4902i q^{75} +52.7542 q^{77} -129.294i q^{79} -33.6974 q^{81} -44.7659i q^{83} +52.7174 q^{85} +55.0081i q^{87} +81.3817 q^{89} +51.2759i q^{91} +22.2685 q^{93} +62.3370i q^{95} +112.525 q^{97} -71.9461i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9}+O(q^{10}) $ 16 * q - 48 * q^9	$ 16 q - 48 q^{9} - 32 q^{21} + 80 q^{25} + 96 q^{29} + 32 q^{33} - 96 q^{41} - 160 q^{45} - 176 q^{49} - 64 q^{53} + 352 q^{57} + 192 q^{61} + 352 q^{69} - 320 q^{73} - 704 q^{77} - 48 q^{81} - 96 q^{89} + 640 q^{93} + 448 q^{97}+O(q^{100}) $ 16 * q - 48 * q^9 - 32 * q^21 + 80 * q^25 + 96 * q^29 + 32 * q^33 - 96 * q^41 - 160 * q^45 - 176 * q^49 - 64 * q^53 + 352 * q^57 + 192 * q^61 + 352 * q^69 - 320 * q^73 - 704 * q^77 - 48 * q^81 - 96 * q^89 + 640 * q^93 + 448 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.29803i	0.766012i	0.923746	+	0.383006i	$0.125111\pi$
$3$	2.29803i	0.766012i	−0.923746	+	0.383006i	$0.874889\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	2.72697i	0.389567i	0.980846	+	0.194784i	$0.0624004\pi$
$7$	2.72697i	0.389567i	−0.980846	+	0.194784i	$0.937600\pi$
$8$	0	0
$9$	3.71904	0.413226
$10$	0	0
$11$	− 19.3454i	− 1.75867i	−0.476204	−	0.879335i	$-0.657988\pi$
$11$	− 19.3454i	− 1.75867i	0.476204	−	0.879335i	$-0.342012\pi$
$12$	0	0
$13$	18.8033	1.44641	0.723203	−	0.690636i	$-0.242669\pi$
$13$	18.8033	1.44641	0.723203	+	0.690636i	$0.242669\pi$
$14$	0	0
$15$	− 5.13856i	− 0.342571i
$16$	0	0
$17$	−23.5759	−1.38682	−0.693410	−	0.720543i	$-0.743892\pi$
$17$	−23.5759	−1.38682	−0.693410	+	0.720543i	$0.743892\pi$
$18$	0	0
$19$	− 27.8780i	− 1.46726i	−0.679549	−	0.733631i	$-0.737824\pi$
$19$	− 27.8780i	− 1.46726i	0.679549	−	0.733631i	$-0.262176\pi$
$20$	0	0
$21$	−6.26667	−0.298413
$22$	0	0
$23$	8.78218i	0.381834i	0.981606	+	0.190917i	$0.0611461\pi$
$23$	8.78218i	0.381834i	−0.981606	+	0.190917i	$0.938854\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	29.2288i	1.08255i
$28$	0	0
$29$	23.9370	0.825414	0.412707	−	0.910864i	$-0.364583\pi$
$29$	23.9370	0.825414	0.412707	+	0.910864i	$0.364583\pi$
$30$	0	0
$31$	− 9.69022i	− 0.312588i	−0.987711	−	0.156294i	$-0.950045\pi$
$31$	− 9.69022i	− 0.312588i	0.987711	−	0.156294i	$-0.0499547\pi$
$32$	0	0
$33$	44.4563	1.34716
$34$	0	0
$35$	− 6.09769i	− 0.174220i
$36$	0	0
$37$	69.6453	1.88231	0.941153	−	0.337982i	$-0.109744\pi$
$37$	69.6453	1.88231	0.941153	+	0.337982i	$0.109744\pi$
$38$	0	0
$39$	43.2106i	1.10796i
$40$	0	0
$41$	32.7276	0.798234	0.399117	−	0.916900i	$-0.369317\pi$
$41$	32.7276	0.798234	0.399117	+	0.916900i	$0.369317\pi$
$42$	0	0
$43$	− 28.2097i	− 0.656040i	−0.944671	−	0.328020i	$-0.893619\pi$
$43$	− 28.2097i	− 0.656040i	0.944671	−	0.328020i	$-0.106381\pi$
$44$	0	0
$45$	−8.31602	−0.184800
$46$	0	0
$47$	37.6454i	0.800966i	0.916304	+	0.400483i	$0.131158\pi$
$47$	37.6454i	0.800966i	−0.916304	+	0.400483i	$0.868842\pi$
$48$	0	0
$49$	41.5636	0.848238
$50$	0	0
$51$	− 54.1783i	− 1.06232i
$52$	0	0
$53$	−42.7329	−0.806281	−0.403140	−	0.915138i	$-0.632081\pi$
$53$	−42.7329	−0.806281	−0.403140	+	0.915138i	$0.632081\pi$
$54$	0	0
$55$	43.2575i	0.786501i
$56$	0	0
$57$	64.0645	1.12394
$58$	0	0
$59$	18.3369i	0.310796i	0.987852	+	0.155398i	$0.0496659\pi$
$59$	18.3369i	0.310796i	−0.987852	+	0.155398i	$0.950334\pi$
$60$	0	0
$61$	35.2430	0.577753	0.288877	−	0.957366i	$-0.406718\pi$
$61$	35.2430	0.577753	0.288877	+	0.957366i	$0.406718\pi$
$62$	0	0
$63$	10.1417i	0.160979i
$64$	0	0
$65$	−42.0454	−0.646852
$66$	0	0
$67$	− 47.0790i	− 0.702671i	−0.936250	−	0.351336i	$-0.885728\pi$
$67$	− 47.0790i	− 0.702671i	0.936250	−	0.351336i	$-0.114272\pi$
$68$	0	0
$69$	−20.1818	−0.292489
$70$	0	0
$71$	85.2253i	1.20036i	0.799866	+	0.600178i	$0.204904\pi$
$71$	85.2253i	1.20036i	−0.799866	+	0.600178i	$0.795096\pi$
$72$	0	0
$73$	15.1258	0.207203	0.103602	−	0.994619i	$-0.466963\pi$
$73$	15.1258	0.207203	0.103602	+	0.994619i	$0.466963\pi$
$74$	0	0
$75$	11.4902i	0.153202i
$76$	0	0
$77$	52.7542	0.685120
$78$	0	0
$79$	− 129.294i	− 1.63664i	−0.574765	−	0.818319i	$-0.694906\pi$
$79$	− 129.294i	− 1.63664i	0.574765	−	0.818319i	$-0.305094\pi$
$80$	0	0
$81$	−33.6974	−0.416018
$82$	0	0
$83$	− 44.7659i	− 0.539348i	−0.962952	−	0.269674i	$-0.913084\pi$
$83$	− 44.7659i	− 0.539348i	0.962952	−	0.269674i	$-0.0869160\pi$
$84$	0	0
$85$	52.7174	0.620205
$86$	0	0
$87$	55.0081i	0.632277i
$88$	0	0
$89$	81.3817	0.914401	0.457201	−	0.889364i	$-0.348852\pi$
$89$	81.3817	0.914401	0.457201	+	0.889364i	$0.348852\pi$
$90$	0	0
$91$	51.2759i	0.563472i
$92$	0	0
$93$	22.2685	0.239446
$94$	0	0
$95$	62.3370i	0.656179i
$96$	0	0
$97$	112.525	1.16006	0.580028	−	0.814597i	$-0.303042\pi$
$97$	112.525	1.16006	0.580028	+	0.814597i	$0.303042\pi$
$98$	0	0
$99$	− 71.9461i	− 0.726728i
$100$	0	0
$101$	−63.6119	−0.629821	−0.314911	−	0.949121i	$-0.601975\pi$
$101$	−63.6119	−0.629821	−0.314911	+	0.949121i	$0.601975\pi$
$102$	0	0
$103$	76.2794i	0.740577i	0.928917	+	0.370289i	$0.120741\pi$
$103$	76.2794i	0.740577i	−0.928917	+	0.370289i	$0.879259\pi$
$104$	0	0
$105$	14.0127	0.133454
$106$	0	0
$107$	111.835i	1.04519i	0.852582	+	0.522594i	$0.175036\pi$
$107$	111.835i	1.04519i	−0.852582	+	0.522594i	$0.824964\pi$
$108$	0	0
$109$	−127.374	−1.16857	−0.584286	−	0.811548i	$-0.698625\pi$
$109$	−127.374	−1.16857	−0.584286	+	0.811548i	$0.698625\pi$
$110$	0	0
$111$	160.047i	1.44187i
$112$	0	0
$113$	15.6077	0.138121	0.0690605	−	0.997612i	$-0.478000\pi$
$113$	15.6077	0.138121	0.0690605	+	0.997612i	$0.478000\pi$
$114$	0	0
$115$	− 19.6376i	− 0.170761i
$116$	0	0
$117$	69.9300	0.597693
$118$	0	0
$119$	− 64.2908i	− 0.540259i
$120$	0	0
$121$	−253.243	−2.09292
$122$	0	0
$123$	75.2092i	0.611457i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 148.087i	− 1.16604i	−0.812459	−	0.583018i	$-0.801872\pi$
$127$	− 148.087i	− 1.16604i	0.812459	−	0.583018i	$-0.198128\pi$
$128$	0	0
$129$	64.8269	0.502534
$130$	0	0
$131$	− 172.818i	− 1.31922i	−0.751606	−	0.659612i	$-0.770721\pi$
$131$	− 172.818i	− 1.31922i	0.751606	−	0.659612i	$-0.229279\pi$
$132$	0	0
$133$	76.0223	0.571597
$134$	0	0
$135$	− 65.3576i	− 0.484130i
$136$	0	0
$137$	−162.370	−1.18518	−0.592591	−	0.805503i	$-0.701895\pi$
$137$	−162.370	−1.18518	−0.592591	+	0.805503i	$0.701895\pi$
$138$	0	0
$139$	− 75.1316i	− 0.540515i	−0.962788	−	0.270257i	$-0.912891\pi$
$139$	− 75.1316i	− 0.540515i	0.962788	−	0.270257i	$-0.0871088\pi$
$140$	0	0
$141$	−86.5105	−0.613550
$142$	0	0
$143$	− 363.756i	− 2.54375i
$144$	0	0
$145$	−53.5248	−0.369136
$146$	0	0
$147$	95.5147i	0.649760i
$148$	0	0
$149$	256.461	1.72121	0.860607	−	0.509270i	$-0.170084\pi$
$149$	256.461	1.72121	0.860607	+	0.509270i	$0.170084\pi$
$150$	0	0
$151$	121.764i	0.806382i	0.915116	+	0.403191i	$0.132099\pi$
$151$	121.764i	0.806382i	−0.915116	+	0.403191i	$0.867901\pi$
$152$	0	0
$153$	−87.6797	−0.573070
$154$	0	0
$155$	21.6680i	0.139793i
$156$	0	0
$157$	97.7472	0.622593	0.311297	−	0.950313i	$-0.399237\pi$
$157$	97.7472	0.622593	0.311297	+	0.950313i	$0.399237\pi$
$158$	0	0
$159$	− 98.2016i	− 0.617620i
$160$	0	0
$161$	−23.9487	−0.148750
$162$	0	0
$163$	− 92.0819i	− 0.564920i	−0.959279	−	0.282460i	$-0.908850\pi$
$163$	− 92.0819i	− 0.564920i	0.959279	−	0.282460i	$-0.0911504\pi$
$164$	0	0
$165$	−99.4073	−0.602469
$166$	0	0
$167$	− 150.795i	− 0.902961i	−0.892281	−	0.451481i	$-0.850896\pi$
$167$	− 150.795i	− 0.902961i	0.892281	−	0.451481i	$-0.149104\pi$
$168$	0	0
$169$	184.563	1.09209
$170$	0	0
$171$	− 103.679i	− 0.606311i
$172$	0	0
$173$	−37.9223	−0.219204	−0.109602	−	0.993976i	$-0.534958\pi$
$173$	−37.9223	−0.219204	−0.109602	+	0.993976i	$0.534958\pi$
$174$	0	0
$175$	13.6348i	0.0779134i
$176$	0	0
$177$	−42.1389	−0.238073
$178$	0	0
$179$	356.798i	1.99328i	0.0818844	+	0.996642i	$0.473906\pi$
$179$	356.798i	1.99328i	−0.0818844	+	0.996642i	$0.526094\pi$
$180$	0	0
$181$	−330.928	−1.82833	−0.914166	−	0.405340i	$-0.867153\pi$
$181$	−330.928	−1.82833	−0.914166	+	0.405340i	$0.867153\pi$
$182$	0	0
$183$	80.9895i	0.442566i
$184$	0	0
$185$	−155.732	−0.841792
$186$	0	0
$187$	456.085i	2.43896i
$188$	0	0
$189$	−79.7060	−0.421725
$190$	0	0
$191$	− 358.323i	− 1.87603i	−0.346588	−	0.938017i	$-0.612660\pi$
$191$	− 358.323i	− 1.87603i	0.346588	−	0.938017i	$-0.387340\pi$
$192$	0	0
$193$	88.3050	0.457539	0.228769	−	0.973481i	$-0.426530\pi$
$193$	88.3050	0.457539	0.228769	+	0.973481i	$0.426530\pi$
$194$	0	0
$195$	− 96.6218i	− 0.495496i
$196$	0	0
$197$	−94.8836	−0.481643	−0.240821	−	0.970569i	$-0.577417\pi$
$197$	−94.8836	−0.481643	−0.240821	+	0.970569i	$0.577417\pi$
$198$	0	0
$199$	119.615i	0.601079i	0.953769	+	0.300539i	$0.0971666\pi$
$199$	119.615i	0.601079i	−0.953769	+	0.300539i	$0.902833\pi$
$200$	0	0
$201$	108.189	0.538254
$202$	0	0
$203$	65.2755i	0.321554i
$204$	0	0
$205$	−73.1811	−0.356981
$206$	0	0
$207$	32.6612i	0.157784i
$208$	0	0
$209$	−539.309	−2.58043
$210$	0	0
$211$	− 85.5156i	− 0.405287i	−0.979253	−	0.202644i	$-0.935047\pi$
$211$	− 85.5156i	− 0.405287i	0.979253	−	0.202644i	$-0.0649533\pi$
$212$	0	0
$213$	−195.851	−0.919487
$214$	0	0
$215$	63.0788i	0.293390i
$216$	0	0
$217$	26.4249	0.121774
$218$	0	0
$219$	34.7597i	0.158720i
$220$	0	0
$221$	−443.305	−2.00590
$222$	0	0
$223$	276.039i	1.23784i	0.785453	+	0.618921i	$0.212430\pi$
$223$	276.039i	1.23784i	−0.785453	+	0.618921i	$0.787570\pi$
$224$	0	0
$225$	18.5952	0.0826452
$226$	0	0
$227$	87.2820i	0.384502i	0.981346	+	0.192251i	$0.0615788\pi$
$227$	87.2820i	0.384502i	−0.981346	+	0.192251i	$0.938421\pi$
$228$	0	0
$229$	99.1232	0.432853	0.216426	−	0.976299i	$-0.430560\pi$
$229$	99.1232	0.432853	0.216426	+	0.976299i	$0.430560\pi$
$230$	0	0
$231$	121.231i	0.524810i
$232$	0	0
$233$	−165.452	−0.710094	−0.355047	−	0.934849i	$-0.615535\pi$
$233$	−165.452	−0.710094	−0.355047	+	0.934849i	$0.615535\pi$
$234$	0	0
$235$	− 84.1777i	− 0.358203i
$236$	0	0
$237$	297.123	1.25368
$238$	0	0
$239$	− 75.3281i	− 0.315180i	−0.987505	−	0.157590i	$-0.949628\pi$
$239$	− 75.3281i	− 0.315180i	0.987505	−	0.157590i	$-0.0503725\pi$
$240$	0	0
$241$	−253.744	−1.05288	−0.526441	−	0.850212i	$-0.676474\pi$
$241$	−253.744	−1.05288	−0.526441	+	0.850212i	$0.676474\pi$
$242$	0	0
$243$	185.621i	0.763873i
$244$	0	0
$245$	−92.9391	−0.379343
$246$	0	0
$247$	− 524.197i	− 2.12225i
$248$	0	0
$249$	102.874	0.413147
$250$	0	0
$251$	− 214.543i	− 0.854753i	−0.904074	−	0.427377i	$-0.859438\pi$
$251$	− 214.543i	− 0.854753i	0.904074	−	0.427377i	$-0.140562\pi$
$252$	0	0
$253$	169.894	0.671520
$254$	0	0
$255$	121.146i	0.475084i
$256$	0	0
$257$	412.080	1.60343	0.801713	−	0.597709i	$-0.203922\pi$
$257$	412.080	1.60343	0.801713	+	0.597709i	$0.203922\pi$
$258$	0	0
$259$	189.921i	0.733284i
$260$	0	0
$261$	89.0226	0.341083
$262$	0	0
$263$	− 339.579i	− 1.29118i	−0.763686	−	0.645588i	$-0.776612\pi$
$263$	− 339.579i	− 1.29118i	0.763686	−	0.645588i	$-0.223388\pi$
$264$	0	0
$265$	95.5536	0.360580
$266$	0	0
$267$	187.018i	0.700442i
$268$	0	0
$269$	−119.714	−0.445034	−0.222517	−	0.974929i	$-0.571427\pi$
$269$	−119.714	−0.445034	−0.222517	+	0.974929i	$0.571427\pi$
$270$	0	0
$271$	404.838i	1.49387i	0.664898	+	0.746934i	$0.268475\pi$
$271$	404.838i	1.49387i	−0.664898	+	0.746934i	$0.731525\pi$
$272$	0	0
$273$	−117.834	−0.431626
$274$	0	0
$275$	− 96.7268i	− 0.351734i
$276$	0	0
$277$	25.1545	0.0908104	0.0454052	−	0.998969i	$-0.485542\pi$
$277$	25.1545	0.0908104	0.0454052	+	0.998969i	$0.485542\pi$
$278$	0	0
$279$	− 36.0383i	− 0.129169i
$280$	0	0
$281$	−454.496	−1.61742	−0.808712	−	0.588205i	$-0.799835\pi$
$281$	−454.496	−1.61742	−0.808712	+	0.588205i	$0.799835\pi$
$282$	0	0
$283$	111.479i	0.393918i	0.980412	+	0.196959i	$0.0631067\pi$
$283$	111.479i	0.393918i	−0.980412	+	0.196959i	$0.936893\pi$
$284$	0	0
$285$	−143.253	−0.502641
$286$	0	0
$287$	89.2472i	0.310966i
$288$	0	0
$289$	266.825	0.923269
$290$	0	0
$291$	258.587i	0.888616i
$292$	0	0
$293$	−310.130	−1.05847	−0.529233	−	0.848477i	$-0.677520\pi$
$293$	−310.130	−1.05847	−0.529233	+	0.848477i	$0.677520\pi$
$294$	0	0
$295$	− 41.0027i	− 0.138992i
$296$	0	0
$297$	565.442	1.90384
$298$	0	0
$299$	165.134i	0.552287i
$300$	0	0
$301$	76.9270	0.255571
$302$	0	0
$303$	− 146.182i	− 0.482450i
$304$	0	0
$305$	−78.8056	−0.258379
$306$	0	0
$307$	− 510.536i	− 1.66298i	−0.555538	−	0.831491i	$-0.687488\pi$
$307$	− 510.536i	− 1.66298i	0.555538	−	0.831491i	$-0.312512\pi$
$308$	0	0
$309$	−175.293	−0.567291
$310$	0	0
$311$	315.393i	1.01413i	0.861909	+	0.507063i	$0.169269\pi$
$311$	315.393i	1.01413i	−0.861909	+	0.507063i	$0.830731\pi$
$312$	0	0
$313$	80.4580	0.257054	0.128527	−	0.991706i	$-0.458975\pi$
$313$	80.4580	0.257054	0.128527	+	0.991706i	$0.458975\pi$
$314$	0	0
$315$	− 22.6775i	− 0.0719921i
$316$	0	0
$317$	−135.504	−0.427458	−0.213729	−	0.976893i	$-0.568561\pi$
$317$	−135.504	−0.427458	−0.213729	+	0.976893i	$0.568561\pi$
$318$	0	0
$319$	− 463.070i	− 1.45163i
$320$	0	0
$321$	−257.001	−0.800626
$322$	0	0
$323$	657.249i	2.03483i
$324$	0	0
$325$	94.0163	0.289281
$326$	0	0
$327$	− 292.710i	− 0.895139i
$328$	0	0
$329$	−102.658	−0.312030
$330$	0	0
$331$	− 435.738i	− 1.31643i	−0.752831	−	0.658214i	$-0.771312\pi$
$331$	− 435.738i	− 1.31643i	0.752831	−	0.658214i	$-0.228688\pi$
$332$	0	0
$333$	259.013	0.777818
$334$	0	0
$335$	105.272i	0.314244i
$336$	0	0
$337$	−66.1157	−0.196189	−0.0980945	−	0.995177i	$-0.531275\pi$
$337$	−66.1157	−0.196189	−0.0980945	+	0.995177i	$0.531275\pi$
$338$	0	0
$339$	35.8670i	0.105802i
$340$	0	0
$341$	−187.461	−0.549738
$342$	0	0
$343$	246.964i	0.720012i
$344$	0	0
$345$	45.1278	0.130805
$346$	0	0
$347$	− 232.394i	− 0.669723i	−0.942267	−	0.334862i	$-0.891310\pi$
$347$	− 232.394i	− 0.669723i	0.942267	−	0.334862i	$-0.108690\pi$
$348$	0	0
$349$	458.147	1.31274	0.656371	−	0.754438i	$-0.272091\pi$
$349$	458.147	1.31274	0.656371	+	0.754438i	$0.272091\pi$
$350$	0	0
$351$	549.597i	1.56580i
$352$	0	0
$353$	376.363	1.06619	0.533093	−	0.846057i	$-0.321030\pi$
$353$	376.363	1.06619	0.533093	+	0.846057i	$0.321030\pi$
$354$	0	0
$355$	− 190.570i	− 0.536816i
$356$	0	0
$357$	147.743	0.413845
$358$	0	0
$359$	193.553i	0.539145i	0.962980	+	0.269572i	$0.0868824\pi$
$359$	193.553i	0.539145i	−0.962980	+	0.269572i	$0.913118\pi$
$360$	0	0
$361$	−416.181	−1.15286
$362$	0	0
$363$	− 581.961i	− 1.60320i
$364$	0	0
$365$	−33.8224	−0.0926641
$366$	0	0
$367$	− 5.10203i	− 0.0139020i	−0.999976	−	0.00695100i	$-0.997787\pi$
$367$	− 5.10203i	− 0.0139020i	0.999976	−	0.00695100i	$-0.00221259\pi$
$368$	0	0
$369$	121.715	0.329851
$370$	0	0
$371$	− 116.531i	− 0.314100i
$372$	0	0
$373$	721.093	1.93322	0.966612	−	0.256244i	$-0.0824849\pi$
$373$	721.093	1.93322	0.966612	+	0.256244i	$0.0824849\pi$
$374$	0	0
$375$	− 25.6928i	− 0.0685142i
$376$	0	0
$377$	450.094	1.19388
$378$	0	0
$379$	− 323.886i	− 0.854580i	−0.904115	−	0.427290i	$-0.859468\pi$
$379$	− 323.886i	− 0.854580i	0.904115	−	0.427290i	$-0.140532\pi$
$380$	0	0
$381$	340.308	0.893197
$382$	0	0
$383$	249.254i	0.650794i	0.945578	+	0.325397i	$0.105498\pi$
$383$	249.254i	0.650794i	−0.945578	+	0.325397i	$0.894502\pi$
$384$	0	0
$385$	−117.962	−0.306395
$386$	0	0
$387$	− 104.913i	− 0.271093i
$388$	0	0
$389$	353.367	0.908398	0.454199	−	0.890900i	$-0.349925\pi$
$389$	353.367	0.908398	0.454199	+	0.890900i	$0.349925\pi$
$390$	0	0
$391$	− 207.048i	− 0.529535i
$392$	0	0
$393$	397.143	1.01054
$394$	0	0
$395$	289.111i	0.731926i
$396$	0	0
$397$	−32.9297	−0.0829463	−0.0414732	−	0.999140i	$-0.513205\pi$
$397$	−32.9297	−0.0829463	−0.0414732	+	0.999140i	$0.513205\pi$
$398$	0	0
$399$	174.702i	0.437850i
$400$	0	0
$401$	−215.009	−0.536183	−0.268092	−	0.963393i	$-0.586393\pi$
$401$	−215.009	−0.536183	−0.268092	+	0.963393i	$0.586393\pi$
$402$	0	0
$403$	− 182.208i	− 0.452128i
$404$	0	0
$405$	75.3498	0.186049
$406$	0	0
$407$	− 1347.31i	− 3.31035i
$408$	0	0
$409$	−114.925	−0.280990	−0.140495	−	0.990081i	$-0.544869\pi$
$409$	−114.925	−0.280990	−0.140495	+	0.990081i	$0.544869\pi$
$410$	0	0
$411$	− 373.132i	− 0.907864i
$412$	0	0
$413$	−50.0043	−0.121076
$414$	0	0
$415$	100.100i	0.241204i
$416$	0	0
$417$	172.655	0.414041
$418$	0	0
$419$	526.123i	1.25566i	0.778349	+	0.627831i	$0.216057\pi$
$419$	526.123i	1.25566i	−0.778349	+	0.627831i	$0.783943\pi$
$420$	0	0
$421$	−298.264	−0.708465	−0.354233	−	0.935157i	$-0.615258\pi$
$421$	−298.264	−0.708465	−0.354233	+	0.935157i	$0.615258\pi$
$422$	0	0
$423$	140.005i	0.330980i
$424$	0	0
$425$	−117.880	−0.277364
$426$	0	0
$427$	96.1065i	0.225074i
$428$	0	0
$429$	835.924	1.94854
$430$	0	0
$431$	319.961i	0.742370i	0.928559	+	0.371185i	$0.121048\pi$
$431$	319.961i	0.742370i	−0.928559	+	0.371185i	$0.878952\pi$
$432$	0	0
$433$	−578.492	−1.33601	−0.668005	−	0.744157i	$-0.732852\pi$
$433$	−578.492	−1.33601	−0.668005	+	0.744157i	$0.732852\pi$
$434$	0	0
$435$	− 123.002i	− 0.282763i
$436$	0	0
$437$	244.829	0.560250
$438$	0	0
$439$	716.216i	1.63147i	0.578425	+	0.815735i	$0.303667\pi$
$439$	716.216i	1.63147i	−0.578425	+	0.815735i	$0.696333\pi$
$440$	0	0
$441$	154.577	0.350514
$442$	0	0
$443$	− 653.956i	− 1.47620i	−0.674692	−	0.738100i	$-0.735723\pi$
$443$	− 653.956i	− 1.47620i	0.674692	−	0.738100i	$-0.264277\pi$
$444$	0	0
$445$	−181.975	−0.408933
$446$	0	0
$447$	589.356i	1.31847i
$448$	0	0
$449$	375.502	0.836307	0.418154	−	0.908376i	$-0.362677\pi$
$449$	375.502	0.836307	0.418154	+	0.908376i	$0.362677\pi$
$450$	0	0
$451$	− 633.127i	− 1.40383i
$452$	0	0
$453$	−279.817	−0.617698
$454$	0	0
$455$	− 114.656i	− 0.251992i
$456$	0	0
$457$	347.650	0.760721	0.380361	−	0.924838i	$-0.375800\pi$
$457$	347.650	0.760721	0.380361	+	0.924838i	$0.375800\pi$
$458$	0	0
$459$	− 689.096i	− 1.50130i
$460$	0	0
$461$	−329.633	−0.715039	−0.357520	−	0.933906i	$-0.616377\pi$
$461$	−329.633	−0.715039	−0.357520	+	0.933906i	$0.616377\pi$
$462$	0	0
$463$	256.443i	0.553872i	0.960888	+	0.276936i	$0.0893190\pi$
$463$	256.443i	0.553872i	−0.960888	+	0.276936i	$0.910681\pi$
$464$	0	0
$465$	−49.7938	−0.107083
$466$	0	0
$467$	595.434i	1.27502i	0.770442	+	0.637510i	$0.220035\pi$
$467$	595.434i	1.27502i	−0.770442	+	0.637510i	$0.779965\pi$
$468$	0	0
$469$	128.383	0.273738
$470$	0	0
$471$	224.626i	0.476914i
$472$	0	0
$473$	−545.727	−1.15376
$474$	0	0
$475$	− 139.390i	− 0.293452i
$476$	0	0
$477$	−158.925	−0.333176
$478$	0	0
$479$	169.276i	0.353395i	0.984265	+	0.176698i	$0.0565414\pi$
$479$	169.276i	0.353395i	−0.984265	+	0.176698i	$0.943459\pi$
$480$	0	0
$481$	1309.56	2.72258
$482$	0	0
$483$	− 55.0350i	− 0.113944i
$484$	0	0
$485$	−251.614	−0.518792
$486$	0	0
$487$	692.389i	1.42174i	0.703321	+	0.710872i	$0.251700\pi$
$487$	692.389i	1.42174i	−0.703321	+	0.710872i	$0.748300\pi$
$488$	0	0
$489$	211.607	0.432735
$490$	0	0
$491$	253.027i	0.515330i	0.966234	+	0.257665i	$0.0829531\pi$
$491$	253.027i	0.515330i	−0.966234	+	0.257665i	$0.917047\pi$
$492$	0	0
$493$	−564.337	−1.14470
$494$	0	0
$495$	160.876i	0.325003i
$496$	0	0
$497$	−232.407	−0.467619
$498$	0	0
$499$	856.886i	1.71721i	0.512640	+	0.858604i	$0.328668\pi$
$499$	856.886i	1.71721i	−0.512640	+	0.858604i	$0.671332\pi$
$500$	0	0
$501$	346.531	0.691679
$502$	0	0
$503$	53.5054i	0.106373i	0.998585	+	0.0531863i	$0.0169377\pi$
$503$	53.5054i	0.106373i	−0.998585	+	0.0531863i	$0.983062\pi$
$504$	0	0
$505$	142.241	0.281665
$506$	0	0
$507$	424.132i	0.836552i
$508$	0	0
$509$	−313.259	−0.615439	−0.307720	−	0.951477i	$-0.599566\pi$
$509$	−313.259	−0.615439	−0.307720	+	0.951477i	$0.599566\pi$
$510$	0	0
$511$	41.2477i	0.0807195i
$512$	0	0
$513$	814.839	1.58838
$514$	0	0
$515$	− 170.566i	− 0.331196i
$516$	0	0
$517$	728.264	1.40864
$518$	0	0
$519$	− 87.1468i	− 0.167913i
$520$	0	0
$521$	291.820	0.560116	0.280058	−	0.959983i	$-0.409646\pi$
$521$	291.820	0.560116	0.280058	+	0.959983i	$0.409646\pi$
$522$	0	0
$523$	236.297i	0.451811i	0.974149	+	0.225906i	$0.0725341\pi$
$523$	236.297i	0.451811i	−0.974149	+	0.225906i	$0.927466\pi$
$524$	0	0
$525$	−31.3334	−0.0596826
$526$	0	0
$527$	228.456i	0.433503i
$528$	0	0
$529$	451.873	0.854203
$530$	0	0
$531$	68.1958i	0.128429i
$532$	0	0
$533$	615.386	1.15457
$534$	0	0
$535$	− 250.071i	− 0.467422i
$536$	0	0
$537$	−819.934	−1.52688
$538$	0	0
$539$	− 804.064i	− 1.49177i
$540$	0	0
$541$	−479.488	−0.886300	−0.443150	−	0.896448i	$-0.646139\pi$
$541$	−479.488	−0.886300	−0.443150	+	0.896448i	$0.646139\pi$
$542$	0	0
$543$	− 760.484i	− 1.40052i
$544$	0	0
$545$	284.817	0.522601
$546$	0	0
$547$	− 606.330i	− 1.10846i	−0.832362	−	0.554232i	$-0.813012\pi$
$547$	− 606.330i	− 1.10846i	0.832362	−	0.554232i	$-0.186988\pi$
$548$	0	0
$549$	131.070	0.238743
$550$	0	0
$551$	− 667.315i	− 1.21110i
$552$	0	0
$553$	352.582	0.637580
$554$	0	0
$555$	− 357.877i	− 0.644823i
$556$	0	0
$557$	22.2604	0.0399649	0.0199824	−	0.999800i	$-0.493639\pi$
$557$	22.2604	0.0399649	0.0199824	+	0.999800i	$0.493639\pi$
$558$	0	0
$559$	− 530.435i	− 0.948899i
$560$	0	0
$561$	−1048.10	−1.86827
$562$	0	0
$563$	518.786i	0.921468i	0.887538	+	0.460734i	$0.152414\pi$
$563$	518.786i	0.921468i	−0.887538	+	0.460734i	$0.847586\pi$
$564$	0	0
$565$	−34.8998	−0.0617696
$566$	0	0
$567$	− 91.8919i	− 0.162067i
$568$	0	0
$569$	−596.499	−1.04833	−0.524164	−	0.851617i	$-0.675622\pi$
$569$	−596.499	−1.04833	−0.524164	+	0.851617i	$0.675622\pi$
$570$	0	0
$571$	− 290.657i	− 0.509031i	−0.967069	−	0.254515i	$-0.918084\pi$
$571$	− 290.657i	− 0.509031i	0.967069	−	0.254515i	$-0.0819160\pi$
$572$	0	0
$573$	823.438	1.43706
$574$	0	0
$575$	43.9109i	0.0763668i
$576$	0	0
$577$	604.262	1.04725	0.523623	−	0.851950i	$-0.324580\pi$
$577$	604.262	1.04725	0.523623	+	0.851950i	$0.324580\pi$
$578$	0	0
$579$	202.928i	0.350480i
$580$	0	0
$581$	122.075	0.210112
$582$	0	0
$583$	826.683i	1.41798i
$584$	0	0
$585$	−156.368	−0.267296
$586$	0	0
$587$	1147.72i	1.95522i	0.210418	+	0.977611i	$0.432517\pi$
$587$	1147.72i	1.95522i	−0.210418	+	0.977611i	$0.567483\pi$
$588$	0	0
$589$	−270.143	−0.458648
$590$	0	0
$591$	− 218.046i	− 0.368944i
$592$	0	0
$593$	−426.623	−0.719432	−0.359716	−	0.933062i	$-0.617126\pi$
$593$	−426.623	−0.719432	−0.359716	+	0.933062i	$0.617126\pi$
$594$	0	0
$595$	143.759i	0.241611i
$596$	0	0
$597$	−274.879	−0.460433
$598$	0	0
$599$	− 602.645i	− 1.00609i	−0.864262	−	0.503043i	$-0.832214\pi$
$599$	− 602.645i	− 1.00609i	0.864262	−	0.503043i	$-0.167786\pi$
$600$	0	0
$601$	804.359	1.33837	0.669184	−	0.743097i	$-0.266644\pi$
$601$	804.359	1.33837	0.669184	+	0.743097i	$0.266644\pi$
$602$	0	0
$603$	− 175.088i	− 0.290362i
$604$	0	0
$605$	566.269	0.935981
$606$	0	0
$607$	1088.91i	1.79392i	0.442115	+	0.896958i	$0.354228\pi$
$607$	1088.91i	1.79392i	−0.442115	+	0.896958i	$0.645772\pi$
$608$	0	0
$609$	−150.005	−0.246314
$610$	0	0
$611$	707.857i	1.15852i
$612$	0	0
$613$	−369.141	−0.602188	−0.301094	−	0.953595i	$-0.597352\pi$
$613$	−369.141	−0.602188	−0.301094	+	0.953595i	$0.597352\pi$
$614$	0	0
$615$	− 168.173i	− 0.273452i
$616$	0	0
$617$	−31.3293	−0.0507768	−0.0253884	−	0.999678i	$-0.508082\pi$
$617$	−31.3293	−0.0507768	−0.0253884	+	0.999678i	$0.508082\pi$
$618$	0	0
$619$	472.149i	0.762761i	0.924418	+	0.381380i	$0.124551\pi$
$619$	472.149i	0.762761i	−0.924418	+	0.381380i	$0.875449\pi$
$620$	0	0
$621$	−256.693	−0.413353
$622$	0	0
$623$	221.925i	0.356221i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 1239.35i	− 1.97664i
$628$	0	0
$629$	−1641.95	−2.61042
$630$	0	0
$631$	1107.24i	1.75474i	0.479815	+	0.877369i	$0.340704\pi$
$631$	1107.24i	1.75474i	−0.479815	+	0.877369i	$0.659296\pi$
$632$	0	0
$633$	196.518	0.310455
$634$	0	0
$635$	331.132i	0.521467i
$636$	0	0
$637$	781.532	1.22690
$638$	0	0
$639$	316.956i	0.496019i
$640$	0	0
$641$	−609.118	−0.950261	−0.475131	−	0.879915i	$-0.657599\pi$
$641$	−609.118	−0.950261	−0.475131	+	0.879915i	$0.657599\pi$
$642$	0	0
$643$	151.870i	0.236190i	0.993002	+	0.118095i	$0.0376788\pi$
$643$	151.870i	0.236190i	−0.993002	+	0.118095i	$0.962321\pi$
$644$	0	0
$645$	−144.957	−0.224740
$646$	0	0
$647$	− 590.497i	− 0.912670i	−0.889808	−	0.456335i	$-0.849162\pi$
$647$	− 590.497i	− 0.912670i	0.889808	−	0.456335i	$-0.150838\pi$
$648$	0	0
$649$	354.735	0.546587
$650$	0	0
$651$	60.7254i	0.0932802i
$652$	0	0
$653$	−356.873	−0.546514	−0.273257	−	0.961941i	$-0.588101\pi$
$653$	−356.873	−0.546514	−0.273257	+	0.961941i	$0.588101\pi$
$654$	0	0
$655$	386.434i	0.589975i
$656$	0	0
$657$	56.2535	0.0856218
$658$	0	0
$659$	− 309.347i	− 0.469419i	−0.972066	−	0.234709i	$-0.924586\pi$
$659$	− 309.347i	− 0.469419i	0.972066	−	0.234709i	$-0.0754138\pi$
$660$	0	0
$661$	−432.538	−0.654369	−0.327185	−	0.944960i	$-0.606100\pi$
$661$	−432.538	−0.654369	−0.327185	+	0.944960i	$0.606100\pi$
$662$	0	0
$663$	− 1018.73i	− 1.53655i
$664$	0	0
$665$	−169.991	−0.255626
$666$	0	0
$667$	210.219i	0.315171i
$668$	0	0
$669$	−634.347	−0.948202
$670$	0	0
$671$	− 681.788i	− 1.01608i
$672$	0	0
$673$	−896.681	−1.33236	−0.666182	−	0.745789i	$-0.732073\pi$
$673$	−896.681	−1.33236	−0.666182	+	0.745789i	$0.732073\pi$
$674$	0	0
$675$	146.144i	0.216510i
$676$	0	0
$677$	−596.363	−0.880890	−0.440445	−	0.897780i	$-0.645179\pi$
$677$	−596.363	−0.880890	−0.440445	+	0.897780i	$0.645179\pi$
$678$	0	0
$679$	306.853i	0.451919i
$680$	0	0
$681$	−200.577	−0.294533
$682$	0	0
$683$	845.172i	1.23744i	0.785612	+	0.618720i	$0.212348\pi$
$683$	845.172i	1.23744i	−0.785612	+	0.618720i	$0.787652\pi$
$684$	0	0
$685$	363.070	0.530030
$686$	0	0
$687$	227.789i	0.331570i
$688$	0	0
$689$	−803.518	−1.16621
$690$	0	0
$691$	− 99.7834i	− 0.144404i	−0.997390	−	0.0722022i	$-0.976997\pi$
$691$	− 99.7834i	− 0.144404i	0.997390	−	0.0722022i	$-0.0230027\pi$
$692$	0	0
$693$	196.195	0.283109
$694$	0	0
$695$	167.999i	0.241726i
$696$	0	0
$697$	−771.584	−1.10701
$698$	0	0
$699$	− 380.214i	− 0.543940i
$700$	0	0
$701$	258.057	0.368127	0.184063	−	0.982914i	$-0.441075\pi$
$701$	258.057	0.368127	0.184063	+	0.982914i	$0.441075\pi$
$702$	0	0
$703$	− 1941.57i	− 2.76183i
$704$	0	0
$705$	193.443	0.274388
$706$	0	0
$707$	− 173.468i	− 0.245358i
$708$	0	0
$709$	463.455	0.653674	0.326837	−	0.945081i	$-0.394017\pi$
$709$	463.455	0.653674	0.326837	+	0.945081i	$0.394017\pi$
$710$	0	0
$711$	− 480.850i	− 0.676301i
$712$	0	0
$713$	85.1012	0.119357
$714$	0	0
$715$	813.383i	1.13760i
$716$	0	0
$717$	173.107	0.241432
$718$	0	0
$719$	− 1254.95i	− 1.74541i	−0.488249	−	0.872704i	$-0.662364\pi$
$719$	− 1254.95i	− 1.74541i	0.488249	−	0.872704i	$-0.337636\pi$
$720$	0	0
$721$	−208.012	−0.288504
$722$	0	0
$723$	− 583.113i	− 0.806519i
$724$	0	0
$725$	119.685	0.165083
$726$	0	0
$727$	− 450.184i	− 0.619236i	−0.950861	−	0.309618i	$-0.899799\pi$
$727$	− 450.184i	− 0.619236i	0.950861	−	0.309618i	$-0.100201\pi$
$728$	0	0
$729$	−729.841	−1.00115
$730$	0	0
$731$	665.070i	0.909808i
$732$	0	0
$733$	−1413.12	−1.92786	−0.963929	−	0.266160i	$-0.914245\pi$
$733$	−1413.12	−1.92786	−0.963929	+	0.266160i	$0.914245\pi$
$734$	0	0
$735$	− 213.577i	− 0.290581i
$736$	0	0
$737$	−910.760	−1.23577
$738$	0	0
$739$	− 331.195i	− 0.448167i	−0.974570	−	0.224083i	$-0.928061\pi$
$739$	− 331.195i	− 0.448167i	0.974570	−	0.224083i	$-0.0719388\pi$
$740$	0	0
$741$	1204.62	1.62567
$742$	0	0
$743$	168.111i	0.226259i	0.993580	+	0.113130i	$0.0360875\pi$
$743$	168.111i	0.226259i	−0.993580	+	0.113130i	$0.963912\pi$
$744$	0	0
$745$	−573.464	−0.769750
$746$	0	0
$747$	− 166.486i	− 0.222873i
$748$	0	0
$749$	−304.971	−0.407171
$750$	0	0
$751$	− 1259.45i	− 1.67703i	−0.544880	−	0.838514i	$-0.683425\pi$
$751$	− 1259.45i	− 1.67703i	0.544880	−	0.838514i	$-0.316575\pi$
$752$	0	0
$753$	493.027	0.654751
$754$	0	0
$755$	− 272.272i	− 0.360625i
$756$	0	0
$757$	−1184.73	−1.56503	−0.782514	−	0.622633i	$-0.786063\pi$
$757$	−1184.73	−1.56503	−0.782514	+	0.622633i	$0.786063\pi$
$758$	0	0
$759$	390.423i	0.514392i
$760$	0	0
$761$	−136.691	−0.179620	−0.0898099	−	0.995959i	$-0.528626\pi$
$761$	−136.691	−0.179620	−0.0898099	+	0.995959i	$0.528626\pi$
$762$	0	0
$763$	− 347.346i	− 0.455237i
$764$	0	0
$765$	196.058	0.256285
$766$	0	0
$767$	344.794i	0.449536i
$768$	0	0
$769$	621.428	0.808099	0.404049	−	0.914737i	$-0.367602\pi$
$769$	621.428	0.808099	0.404049	+	0.914737i	$0.367602\pi$
$770$	0	0
$771$	946.975i	1.22824i
$772$	0	0
$773$	−209.531	−0.271063	−0.135531	−	0.990773i	$-0.543274\pi$
$773$	−209.531	−0.271063	−0.135531	+	0.990773i	$0.543274\pi$
$774$	0	0
$775$	− 48.4511i	− 0.0625175i
$776$	0	0
$777$	−436.444	−0.561704
$778$	0	0
$779$	− 912.379i	− 1.17122i
$780$	0	0
$781$	1648.71	2.11103
$782$	0	0
$783$	699.650i	0.893550i
$784$	0	0
$785$	−218.569	−0.278432
$786$	0	0
$787$	− 176.067i	− 0.223719i	−0.993724	−	0.111859i	$-0.964319\pi$
$787$	− 176.067i	− 0.223719i	0.993724	−	0.111859i	$-0.0356806\pi$
$788$	0	0
$789$	780.365	0.989056
$790$	0	0
$791$	42.5617i	0.0538074i
$792$	0	0
$793$	662.683	0.835666
$794$	0	0
$795$	219.586i	0.276208i
$796$	0	0
$797$	−1185.07	−1.48692	−0.743458	−	0.668783i	$-0.766816\pi$
$797$	−1185.07	−1.48692	−0.743458	+	0.668783i	$0.766816\pi$
$798$	0	0
$799$	− 887.526i	− 1.11080i
$800$	0	0
$801$	302.662	0.377855
$802$	0	0
$803$	− 292.615i	− 0.364402i
$804$	0	0
$805$	53.5510	0.0665230
$806$	0	0
$807$	− 275.107i	− 0.340901i
$808$	0	0
$809$	537.511	0.664414	0.332207	−	0.943206i	$-0.392207\pi$
$809$	537.511	0.664414	0.332207	+	0.943206i	$0.392207\pi$
$810$	0	0
$811$	859.330i	1.05959i	0.848125	+	0.529797i	$0.177732\pi$
$811$	859.330i	1.05959i	−0.848125	+	0.529797i	$0.822268\pi$
$812$	0	0
$813$	−930.332	−1.14432
$814$	0	0
$815$	205.901i	0.252640i
$816$	0	0
$817$	−786.429	−0.962581
$818$	0	0
$819$	190.697i	0.232841i
$820$	0	0
$821$	−808.044	−0.984220	−0.492110	−	0.870533i	$-0.663774\pi$
$821$	−808.044	−0.984220	−0.492110	+	0.870533i	$0.663774\pi$
$822$	0	0
$823$	709.287i	0.861831i	0.902392	+	0.430916i	$0.141809\pi$
$823$	709.287i	0.861831i	−0.902392	+	0.430916i	$0.858191\pi$
$824$	0	0
$825$	222.282	0.269432
$826$	0	0
$827$	714.799i	0.864328i	0.901795	+	0.432164i	$0.142250\pi$
$827$	714.799i	0.864328i	−0.901795	+	0.432164i	$0.857750\pi$
$828$	0	0
$829$	730.675	0.881393	0.440697	−	0.897656i	$-0.354731\pi$
$829$	730.675	0.881393	0.440697	+	0.897656i	$0.354731\pi$
$830$	0	0
$831$	57.8059i	0.0695618i
$832$	0	0
$833$	−979.902	−1.17635
$834$	0	0
$835$	337.187i	0.403817i
$836$	0	0
$837$	283.233	0.338391
$838$	0	0
$839$	1154.02i	1.37548i	0.725959	+	0.687738i	$0.241396\pi$
$839$	1154.02i	1.37548i	−0.725959	+	0.687738i	$0.758604\pi$
$840$	0	0
$841$	−268.019	−0.318691
$842$	0	0
$843$	− 1044.45i	− 1.23897i
$844$	0	0
$845$	−412.695	−0.488397
$846$	0	0
$847$	− 690.586i	− 0.815332i
$848$	0	0
$849$	−256.182	−0.301746
$850$	0	0
$851$	611.638i	0.718728i
$852$	0	0
$853$	105.220	0.123353	0.0616764	−	0.998096i	$-0.480355\pi$
$853$	105.220	0.123353	0.0616764	+	0.998096i	$0.480355\pi$
$854$	0	0
$855$	231.834i	0.271150i
$856$	0	0
$857$	−681.033	−0.794671	−0.397336	−	0.917673i	$-0.630065\pi$
$857$	−681.033	−0.794671	−0.397336	+	0.917673i	$0.630065\pi$
$858$	0	0
$859$	921.542i	1.07281i	0.843961	+	0.536404i	$0.180218\pi$
$859$	921.542i	1.07281i	−0.843961	+	0.536404i	$0.819782\pi$
$860$	0	0
$861$	−205.093	−0.238203
$862$	0	0
$863$	744.099i	0.862223i	0.902299	+	0.431112i	$0.141878\pi$
$863$	744.099i	0.862223i	−0.902299	+	0.431112i	$0.858122\pi$
$864$	0	0
$865$	84.7969	0.0980311
$866$	0	0
$867$	613.172i	0.707235i
$868$	0	0
$869$	−2501.25	−2.87830
$870$	0	0
$871$	− 885.238i	− 1.01635i
$872$	0	0
$873$	418.486	0.479365
$874$	0	0
$875$	− 30.4884i	− 0.0348439i
$876$	0	0
$877$	−795.922	−0.907551	−0.453775	−	0.891116i	$-0.649923\pi$
$877$	−795.922	−0.907551	−0.453775	+	0.891116i	$0.649923\pi$
$878$	0	0
$879$	− 712.690i	− 0.810797i
$880$	0	0
$881$	1211.69	1.37536	0.687679	−	0.726015i	$-0.258630\pi$
$881$	1211.69	1.37536	0.687679	+	0.726015i	$0.258630\pi$
$882$	0	0
$883$	217.973i	0.246855i	0.992354	+	0.123427i	$0.0393886\pi$
$883$	217.973i	0.246855i	−0.992354	+	0.123427i	$0.960611\pi$
$884$	0	0
$885$	94.2255	0.106470
$886$	0	0
$887$	1236.92i	1.39450i	0.716829	+	0.697249i	$0.245593\pi$
$887$	1236.92i	1.39450i	−0.716829	+	0.697249i	$0.754407\pi$
$888$	0	0
$889$	403.827	0.454249
$890$	0	0
$891$	651.889i	0.731638i
$892$	0	0
$893$	1049.48	1.17523
$894$	0	0
$895$	− 797.824i	− 0.891424i
$896$	0	0
$897$	−379.483	−0.423058
$898$	0	0
$899$	− 231.955i	− 0.258014i
$900$	0	0
$901$	1007.47	1.11817
$902$	0	0
$903$	176.781i	0.195771i
$904$	0	0
$905$	739.978	0.817655
$906$	0	0
$907$	804.548i	0.887043i	0.896264	+	0.443522i	$0.146271\pi$
$907$	804.548i	0.887043i	−0.896264	+	0.443522i	$0.853729\pi$
$908$	0	0
$909$	−236.575	−0.260259
$910$	0	0
$911$	− 761.825i	− 0.836251i	−0.908389	−	0.418126i	$-0.862687\pi$
$911$	− 761.825i	− 0.836251i	0.908389	−	0.418126i	$-0.137313\pi$
$912$	0	0
$913$	−866.013	−0.948535
$914$	0	0
$915$	− 181.098i	− 0.197921i
$916$	0	0
$917$	471.270	0.513926
$918$	0	0
$919$	502.728i	0.547039i	0.961867	+	0.273519i	$0.0881877\pi$
$919$	502.728i	0.547039i	−0.961867	+	0.273519i	$0.911812\pi$
$920$	0	0
$921$	1173.23	1.27386
$922$	0	0
$923$	1602.51i	1.73620i
$924$	0	0
$925$	348.226	0.376461
$926$	0	0
$927$	283.686i	0.306026i
$928$	0	0
$929$	845.520	0.910140	0.455070	−	0.890456i	$-0.349614\pi$
$929$	845.520	0.910140	0.455070	+	0.890456i	$0.349614\pi$
$930$	0	0
$931$	− 1158.71i	− 1.24459i
$932$	0	0
$933$	−724.784	−0.776832
$934$	0	0
$935$	− 1019.84i	− 1.09073i
$936$	0	0
$937$	93.6571	0.0999542	0.0499771	−	0.998750i	$-0.484085\pi$
$937$	93.6571	0.0999542	0.0499771	+	0.998750i	$0.484085\pi$
$938$	0	0
$939$	184.895i	0.196907i
$940$	0	0
$941$	93.5311	0.0993954	0.0496977	−	0.998764i	$-0.484174\pi$
$941$	93.5311	0.0993954	0.0496977	+	0.998764i	$0.484174\pi$
$942$	0	0
$943$	287.420i	0.304793i
$944$	0	0
$945$	178.228	0.188601
$946$	0	0
$947$	− 1061.81i	− 1.12124i	−0.828074	−	0.560619i	$-0.810563\pi$
$947$	− 1061.81i	− 1.12124i	0.828074	−	0.560619i	$-0.189437\pi$
$948$	0	0
$949$	284.415	0.299700
$950$	0	0
$951$	− 311.393i	− 0.327438i
$952$	0	0
$953$	−435.168	−0.456630	−0.228315	−	0.973587i	$-0.573322\pi$
$953$	−435.168	−0.456630	−0.228315	+	0.973587i	$0.573322\pi$
$954$	0	0
$955$	801.234i	0.838988i
$956$	0	0
$957$	1064.15	1.11197
$958$	0	0
$959$	− 442.778i	− 0.461708i
$960$	0	0
$961$	867.100	0.902289
$962$	0	0
$963$	415.919i	0.431899i
$964$	0	0
$965$	−197.456	−0.204618
$966$	0	0
$967$	1400.02i	1.44780i	0.689905	+	0.723900i	$0.257652\pi$
$967$	1400.02i	1.44780i	−0.689905	+	0.723900i	$0.742348\pi$
$968$	0	0
$969$	−1510.38	−1.55870
$970$	0	0
$971$	− 562.184i	− 0.578974i	−0.957182	−	0.289487i	$-0.906515\pi$
$971$	− 562.184i	− 0.578974i	0.957182	−	0.289487i	$-0.0934847\pi$
$972$	0	0
$973$	204.881	0.210567
$974$	0	0
$975$	216.053i	0.221593i
$976$	0	0
$977$	−760.867	−0.778779	−0.389390	−	0.921073i	$-0.627314\pi$
$977$	−760.867	−0.778779	−0.389390	+	0.921073i	$0.627314\pi$
$978$	0	0
$979$	− 1574.36i	− 1.60813i
$980$	0	0
$981$	−473.709	−0.482884
$982$	0	0
$983$	− 239.801i	− 0.243948i	−0.992533	−	0.121974i	$-0.961078\pi$
$983$	− 239.801i	− 0.243948i	0.992533	−	0.121974i	$-0.0389224\pi$
$984$	0	0
$985$	212.166	0.215397
$986$	0	0
$987$	− 235.911i	− 0.239019i
$988$	0	0
$989$	247.743	0.250498
$990$	0	0
$991$	12.0827i	0.0121925i	0.999981	+	0.00609623i	$0.00194050\pi$
$991$	12.0827i	0.0121925i	−0.999981	+	0.00609623i	$0.998059\pi$
$992$	0	0
$993$	1001.34	1.00840
$994$	0	0
$995$	− 267.466i	− 0.268810i
$996$	0	0
$997$	−127.529	−0.127913	−0.0639565	−	0.997953i	$-0.520372\pi$
$997$	−127.529	−0.127913	−0.0639565	+	0.997953i	$0.520372\pi$
$998$	0	0
$999$	2035.65i	2.03769i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.3.b.a.511.11	yes	16
4.3	odd	2	inner	640.3.b.a.511.6	✓	16
8.3	odd	2		640.3.b.b.511.11	yes	16
8.5	even	2		640.3.b.b.511.6	yes	16
16.3	odd	4		1280.3.g.h.1151.10		16
16.5	even	4		1280.3.g.h.1151.9		16
16.11	odd	4		1280.3.g.g.1151.7		16
16.13	even	4		1280.3.g.g.1151.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.b.a.511.6	✓	16	4.3	odd	2	inner
640.3.b.a.511.11	yes	16	1.1	even	1	trivial
640.3.b.b.511.6	yes	16	8.5	even	2
640.3.b.b.511.11	yes	16	8.3	odd	2
1280.3.g.g.1151.7		16	16.11	odd	4
1280.3.g.g.1151.8		16	16.13	even	4
1280.3.g.h.1151.9		16	16.5	even	4
1280.3.g.h.1151.10		16	16.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	640.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.29803i q^{3} -2.23607 q^{5} +2.72697i q^{7} +3.71904 q^{9} +O(q^{10})\)	\(q+2.29803i q^{3} -2.23607 q^{5} +2.72697i q^{7} +3.71904 q^{9} -19.3454i q^{11} +18.8033 q^{13} -5.13856i q^{15} -23.5759 q^{17} -27.8780i q^{19} -6.26667 q^{21} +8.78218i q^{23} +5.00000 q^{25} +29.2288i q^{27} +23.9370 q^{29} -9.69022i q^{31} +44.4563 q^{33} -6.09769i q^{35} +69.6453 q^{37} +43.2106i q^{39} +32.7276 q^{41} -28.2097i q^{43} -8.31602 q^{45} +37.6454i q^{47} +41.5636 q^{49} -54.1783i q^{51} -42.7329 q^{53} +43.2575i q^{55} +64.0645 q^{57} +18.3369i q^{59} +35.2430 q^{61} +10.1417i q^{63} -42.0454 q^{65} -47.0790i q^{67} -20.1818 q^{69} +85.2253i q^{71} +15.1258 q^{73} +11.4902i q^{75} +52.7542 q^{77} -129.294i q^{79} -33.6974 q^{81} -44.7659i q^{83} +52.7174 q^{85} +55.0081i q^{87} +81.3817 q^{89} +51.2759i q^{91} +22.2685 q^{93} +62.3370i q^{95} +112.525 q^{97} -71.9461i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 16 * q - 48 * q^9	\( 16 q - 48 q^{9} - 32 q^{21} + 80 q^{25} + 96 q^{29} + 32 q^{33} - 96 q^{41} - 160 q^{45} - 176 q^{49} - 64 q^{53} + 352 q^{57} + 192 q^{61} + 352 q^{69} - 320 q^{73} - 704 q^{77} - 48 q^{81} - 96 q^{89} + 640 q^{93} + 448 q^{97}+O(q^{100}) \) 16 * q - 48 * q^9 - 32 * q^21 + 80 * q^25 + 96 * q^29 + 32 * q^33 - 96 * q^41 - 160 * q^45 - 176 * q^49 - 64 * q^53 + 352 * q^57 + 192 * q^61 + 352 * q^69 - 320 * q^73 - 704 * q^77 - 48 * q^81 - 96 * q^89 + 640 * q^93 + 448 * q^97

Introduction

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