LMFDB - Embedded newform 1280.3.g.g.1151.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1151,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1280.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$34.8774738381$
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^{36} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.8
Root			$1.56323i$ of defining polynomial
Character	$\chi$	$=$	1280.1151
Dual form			1280.3.g.g.1151.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29803 q^{3} +2.23607i q^{5} -2.72697i q^{7} -3.71904 q^{9} +O(q^{10})$	$q-2.29803 q^{3} +2.23607i q^{5} -2.72697i q^{7} -3.71904 q^{9} -19.3454 q^{11} +18.8033i q^{13} -5.13856i q^{15} -23.5759 q^{17} +27.8780 q^{19} +6.26667i q^{21} -8.78218i q^{23} -5.00000 q^{25} +29.2288 q^{27} +23.9370i q^{29} -9.69022i q^{31} +44.4563 q^{33} +6.09769 q^{35} -69.6453i q^{37} -43.2106i q^{39} -32.7276 q^{41} -28.2097 q^{43} -8.31602i q^{45} +37.6454i q^{47} +41.5636 q^{49} +54.1783 q^{51} +42.7329i q^{53} -43.2575i q^{55} -64.0645 q^{57} +18.3369 q^{59} +35.2430i q^{61} +10.1417i q^{63} -42.0454 q^{65} +47.0790 q^{67} +20.1818i q^{69} -85.2253i q^{71} -15.1258 q^{73} +11.4902 q^{75} +52.7542i q^{77} -129.294i q^{79} -33.6974 q^{81} +44.7659 q^{83} -52.7174i q^{85} -55.0081i q^{87} -81.3817 q^{89} +51.2759 q^{91} +22.2685i q^{93} +62.3370i q^{95} +112.525 q^{97} +71.9461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 48 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 48 * q^9	$ 16 q - 16 q^{3} + 48 q^{9} - 64 q^{11} - 96 q^{19} - 80 q^{25} - 64 q^{27} + 32 q^{33} + 80 q^{35} + 96 q^{41} + 176 q^{43} - 176 q^{49} - 96 q^{51} - 352 q^{57} + 32 q^{59} + 560 q^{67} + 320 q^{73} + 80 q^{75} - 48 q^{81} - 48 q^{83} + 96 q^{89} + 32 q^{91} + 448 q^{97} - 704 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 48 * q^9 - 64 * q^11 - 96 * q^19 - 80 * q^25 - 64 * q^27 + 32 * q^33 + 80 * q^35 + 96 * q^41 + 176 * q^43 - 176 * q^49 - 96 * q^51 - 352 * q^57 + 32 * q^59 + 560 * q^67 + 320 * q^73 + 80 * q^75 - 48 * q^81 - 48 * q^83 + 96 * q^89 + 32 * q^91 + 448 * q^97 - 704 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

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

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
$3$	−2.29803	−0.766012	−0.383006	−	0.923746i	$-0.625111\pi$
$3$	−2.29803	−0.766012	−0.383006	+	0.923746i	$0.625111\pi$
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	− 2.72697i	− 0.389567i	−0.980846	−	0.194784i	$-0.937600\pi$
$7$	− 2.72697i	− 0.389567i	0.980846	−	0.194784i	$-0.0624004\pi$
$8$	0	0
$9$	−3.71904	−0.413226
$10$	0	0
$11$	−19.3454	−1.75867	−0.879335	−	0.476204i	$-0.842012\pi$
$11$	−19.3454	−1.75867	−0.879335	+	0.476204i	$0.842012\pi$
$12$	0	0
$13$	18.8033i	1.44641i	0.690636	+	0.723203i	$0.257331\pi$
$13$	18.8033i	1.44641i	−0.690636	+	0.723203i	$0.742669\pi$
$14$	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
$19$	27.8780	1.46726	0.733631	−	0.679549i	$-0.237824\pi$
$19$	27.8780	1.46726	0.733631	+	0.679549i	$0.237824\pi$
$20$	0	0
$21$	6.26667i	0.298413i
$22$	0	0
$23$	− 8.78218i	− 0.381834i	−0.981606	−	0.190917i	$-0.938854\pi$
$23$	− 8.78218i	− 0.381834i	0.981606	−	0.190917i	$-0.0611461\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	29.2288	1.08255
$28$	0	0
$29$	23.9370i	0.825414i	0.910864	+	0.412707i	$0.135417\pi$
$29$	23.9370i	0.825414i	−0.910864	+	0.412707i	$0.864583\pi$
$30$	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
$35$	6.09769	0.174220
$36$	0	0
$37$	− 69.6453i	− 1.88231i	−0.337982	−	0.941153i	$-0.609744\pi$
$37$	− 69.6453i	− 1.88231i	0.337982	−	0.941153i	$-0.390256\pi$
$38$	0	0
$39$	− 43.2106i	− 1.10796i
$40$	0	0
$41$	−32.7276	−0.798234	−0.399117	−	0.916900i	$-0.630683\pi$
$41$	−32.7276	−0.798234	−0.399117	+	0.916900i	$0.630683\pi$
$42$	0	0
$43$	−28.2097	−0.656040	−0.328020	−	0.944671i	$-0.606381\pi$
$43$	−28.2097	−0.656040	−0.328020	+	0.944671i	$0.606381\pi$
$44$	0	0
$45$	− 8.31602i	− 0.184800i
$46$	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
$51$	54.1783	1.06232
$52$	0	0
$53$	42.7329i	0.806281i	0.915138	+	0.403140i	$0.132081\pi$
$53$	42.7329i	0.806281i	−0.915138	+	0.403140i	$0.867919\pi$
$54$	0	0
$55$	− 43.2575i	− 0.786501i
$56$	0	0
$57$	−64.0645	−1.12394
$58$	0	0
$59$	18.3369	0.310796	0.155398	−	0.987852i	$-0.450334\pi$
$59$	18.3369	0.310796	0.155398	+	0.987852i	$0.450334\pi$
$60$	0	0
$61$	35.2430i	0.577753i	0.957366	+	0.288877i	$0.0932818\pi$
$61$	35.2430i	0.577753i	−0.957366	+	0.288877i	$0.906718\pi$
$62$	0	0
$63$	10.1417i	0.160979i
$64$	0	0
$65$	−42.0454	−0.646852
$66$	0	0
$67$	47.0790	0.702671	0.351336	−	0.936250i	$-0.385728\pi$
$67$	47.0790	0.702671	0.351336	+	0.936250i	$0.385728\pi$
$68$	0	0
$69$	20.1818i	0.292489i
$70$	0	0
$71$	− 85.2253i	− 1.20036i	−0.799866	−	0.600178i	$-0.795096\pi$
$71$	− 85.2253i	− 1.20036i	0.799866	−	0.600178i	$-0.204904\pi$
$72$	0	0
$73$	−15.1258	−0.207203	−0.103602	−	0.994619i	$-0.533037\pi$
$73$	−15.1258	−0.207203	−0.103602	+	0.994619i	$0.533037\pi$
$74$	0	0
$75$	11.4902	0.153202
$76$	0	0
$77$	52.7542i	0.685120i
$78$	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
$83$	44.7659	0.539348	0.269674	−	0.962952i	$-0.413084\pi$
$83$	44.7659	0.539348	0.269674	+	0.962952i	$0.413084\pi$
$84$	0	0
$85$	− 52.7174i	− 0.620205i
$86$	0	0
$87$	− 55.0081i	− 0.632277i
$88$	0	0
$89$	−81.3817	−0.914401	−0.457201	−	0.889364i	$-0.651148\pi$
$89$	−81.3817	−0.914401	−0.457201	+	0.889364i	$0.651148\pi$
$90$	0	0
$91$	51.2759	0.563472
$92$	0	0
$93$	22.2685i	0.239446i
$94$	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
$99$	71.9461	0.726728
$100$	0	0
$101$	63.6119i	0.629821i	0.949121	+	0.314911i	$0.101975\pi$
$101$	63.6119i	0.629821i	−0.949121	+	0.314911i	$0.898025\pi$
$102$	0	0
$103$	− 76.2794i	− 0.740577i	−0.928917	−	0.370289i	$-0.879259\pi$
$103$	− 76.2794i	− 0.740577i	0.928917	−	0.370289i	$-0.120741\pi$
$104$	0	0
$105$	−14.0127	−0.133454
$106$	0	0
$107$	111.835	1.04519	0.522594	−	0.852582i	$-0.324964\pi$
$107$	111.835	1.04519	0.522594	+	0.852582i	$0.324964\pi$
$108$	0	0
$109$	− 127.374i	− 1.16857i	−0.811548	−	0.584286i	$-0.801375\pi$
$109$	− 127.374i	− 1.16857i	0.811548	−	0.584286i	$-0.198625\pi$
$110$	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
$115$	19.6376	0.170761
$116$	0	0
$117$	− 69.9300i	− 0.597693i
$118$	0	0
$119$	64.2908i	0.540259i
$120$	0	0
$121$	253.243	2.09292
$122$	0	0
$123$	75.2092	0.611457
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	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
$131$	172.818	1.31922	0.659612	−	0.751606i	$-0.270721\pi$
$131$	172.818	1.31922	0.659612	+	0.751606i	$0.270721\pi$
$132$	0	0
$133$	− 76.0223i	− 0.571597i
$134$	0	0
$135$	65.3576i	0.484130i
$136$	0	0
$137$	162.370	1.18518	0.592591	−	0.805503i	$-0.298105\pi$
$137$	162.370	1.18518	0.592591	+	0.805503i	$0.298105\pi$
$138$	0	0
$139$	−75.1316	−0.540515	−0.270257	−	0.962788i	$-0.587109\pi$
$139$	−75.1316	−0.540515	−0.270257	+	0.962788i	$0.587109\pi$
$140$	0	0
$141$	− 86.5105i	− 0.613550i
$142$	0	0
$143$	− 363.756i	− 2.54375i
$144$	0	0
$145$	−53.5248	−0.369136
$146$	0	0
$147$	−95.5147	−0.649760
$148$	0	0
$149$	− 256.461i	− 1.72121i	−0.509270	−	0.860607i	$-0.670084\pi$
$149$	− 256.461i	− 1.72121i	0.509270	−	0.860607i	$-0.329916\pi$
$150$	0	0
$151$	− 121.764i	− 0.806382i	−0.915116	−	0.403191i	$-0.867901\pi$
$151$	− 121.764i	− 0.806382i	0.915116	−	0.403191i	$-0.132099\pi$
$152$	0	0
$153$	87.6797	0.573070
$154$	0	0
$155$	21.6680	0.139793
$156$	0	0
$157$	97.7472i	0.622593i	0.950313	+	0.311297i	$0.100763\pi$
$157$	97.7472i	0.622593i	−0.950313	+	0.311297i	$0.899237\pi$
$158$	0	0
$159$	− 98.2016i	− 0.617620i
$160$	0	0
$161$	−23.9487	−0.148750
$162$	0	0
$163$	92.0819	0.564920	0.282460	−	0.959279i	$-0.408850\pi$
$163$	92.0819	0.564920	0.282460	+	0.959279i	$0.408850\pi$
$164$	0	0
$165$	99.4073i	0.602469i
$166$	0	0
$167$	150.795i	0.902961i	0.892281	+	0.451481i	$0.149104\pi$
$167$	150.795i	0.902961i	−0.892281	+	0.451481i	$0.850896\pi$
$168$	0	0
$169$	−184.563	−1.09209
$170$	0	0
$171$	−103.679	−0.606311
$172$	0	0
$173$	− 37.9223i	− 0.219204i	−0.993976	−	0.109602i	$-0.965042\pi$
$173$	− 37.9223i	− 0.219204i	0.993976	−	0.109602i	$-0.0349577\pi$
$174$	0	0
$175$	13.6348i	0.0779134i
$176$	0	0
$177$	−42.1389	−0.238073
$178$	0	0
$179$	−356.798	−1.99328	−0.996642	−	0.0818844i	$-0.973906\pi$
$179$	−356.798	−1.99328	−0.996642	+	0.0818844i	$0.973906\pi$
$180$	0	0
$181$	330.928i	1.82833i	0.405340	+	0.914166i	$0.367153\pi$
$181$	330.928i	1.82833i	−0.405340	+	0.914166i	$0.632847\pi$
$182$	0	0
$183$	− 80.9895i	− 0.442566i
$184$	0	0
$185$	155.732	0.841792
$186$	0	0
$187$	456.085	2.43896
$188$	0	0
$189$	− 79.7060i	− 0.421725i
$190$	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
$195$	96.6218	0.495496
$196$	0	0
$197$	94.8836i	0.481643i	0.970569	+	0.240821i	$0.0774168\pi$
$197$	94.8836i	0.481643i	−0.970569	+	0.240821i	$0.922583\pi$
$198$	0	0
$199$	− 119.615i	− 0.601079i	−0.953769	−	0.300539i	$-0.902833\pi$
$199$	− 119.615i	− 0.601079i	0.953769	−	0.300539i	$-0.0971666\pi$
$200$	0	0
$201$	−108.189	−0.538254
$202$	0	0
$203$	65.2755	0.321554
$204$	0	0
$205$	− 73.1811i	− 0.356981i
$206$	0	0
$207$	32.6612i	0.157784i
$208$	0	0
$209$	−539.309	−2.58043
$210$	0	0
$211$	85.5156	0.405287	0.202644	−	0.979253i	$-0.435047\pi$
$211$	85.5156	0.405287	0.202644	+	0.979253i	$0.435047\pi$
$212$	0	0
$213$	195.851i	0.919487i
$214$	0	0
$215$	− 63.0788i	− 0.293390i
$216$	0	0
$217$	−26.4249	−0.121774
$218$	0	0
$219$	34.7597	0.158720
$220$	0	0
$221$	− 443.305i	− 2.00590i
$222$	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
$227$	−87.2820	−0.384502	−0.192251	−	0.981346i	$-0.561579\pi$
$227$	−87.2820	−0.384502	−0.192251	+	0.981346i	$0.561579\pi$
$228$	0	0
$229$	− 99.1232i	− 0.432853i	−0.976299	−	0.216426i	$-0.930560\pi$
$229$	− 99.1232i	− 0.432853i	0.976299	−	0.216426i	$-0.0694401\pi$
$230$	0	0
$231$	− 121.231i	− 0.524810i
$232$	0	0
$233$	165.452	0.710094	0.355047	−	0.934849i	$-0.384465\pi$
$233$	165.452	0.710094	0.355047	+	0.934849i	$0.384465\pi$
$234$	0	0
$235$	−84.1777	−0.358203
$236$	0	0
$237$	297.123i	1.25368i
$238$	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
$243$	−185.621	−0.763873
$244$	0	0
$245$	92.9391i	0.379343i
$246$	0	0
$247$	524.197i	2.12225i
$248$	0	0
$249$	−102.874	−0.413147
$250$	0	0
$251$	−214.543	−0.854753	−0.427377	−	0.904074i	$-0.640562\pi$
$251$	−214.543	−0.854753	−0.427377	+	0.904074i	$0.640562\pi$
$252$	0	0
$253$	169.894i	0.671520i
$254$	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
$259$	−189.921	−0.733284
$260$	0	0
$261$	− 89.0226i	− 0.341083i
$262$	0	0
$263$	339.579i	1.29118i	0.763686	+	0.645588i	$0.223388\pi$
$263$	339.579i	1.29118i	−0.763686	+	0.645588i	$0.776612\pi$
$264$	0	0
$265$	−95.5536	−0.360580
$266$	0	0
$267$	187.018	0.700442
$268$	0	0
$269$	− 119.714i	− 0.445034i	−0.974929	−	0.222517i	$-0.928573\pi$
$269$	− 119.714i	− 0.445034i	0.974929	−	0.222517i	$-0.0714273\pi$
$270$	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
$275$	96.7268	0.351734
$276$	0	0
$277$	− 25.1545i	− 0.0908104i	−0.998969	−	0.0454052i	$-0.985542\pi$
$277$	− 25.1545i	− 0.0908104i	0.998969	−	0.0454052i	$-0.0144579\pi$
$278$	0	0
$279$	36.0383i	0.129169i
$280$	0	0
$281$	454.496	1.61742	0.808712	−	0.588205i	$-0.200165\pi$
$281$	454.496	1.61742	0.808712	+	0.588205i	$0.200165\pi$
$282$	0	0
$283$	111.479	0.393918	0.196959	−	0.980412i	$-0.436893\pi$
$283$	111.479	0.393918	0.196959	+	0.980412i	$0.436893\pi$
$284$	0	0
$285$	− 143.253i	− 0.502641i
$286$	0	0
$287$	89.2472i	0.310966i
$288$	0	0
$289$	266.825	0.923269
$290$	0	0
$291$	−258.587	−0.888616
$292$	0	0
$293$	310.130i	1.05847i	0.848477	+	0.529233i	$0.177520\pi$
$293$	310.130i	1.05847i	−0.848477	+	0.529233i	$0.822480\pi$
$294$	0	0
$295$	41.0027i	0.138992i
$296$	0	0
$297$	−565.442	−1.90384
$298$	0	0
$299$	165.134	0.552287
$300$	0	0
$301$	76.9270i	0.255571i
$302$	0	0
$303$	− 146.182i	− 0.482450i
$304$	0	0
$305$	−78.8056	−0.258379
$306$	0	0
$307$	510.536	1.66298	0.831491	−	0.555538i	$-0.187488\pi$
$307$	510.536	1.66298	0.831491	+	0.555538i	$0.187488\pi$
$308$	0	0
$309$	175.293i	0.567291i
$310$	0	0
$311$	− 315.393i	− 1.01413i	−0.861909	−	0.507063i	$-0.830731\pi$
$311$	− 315.393i	− 1.01413i	0.861909	−	0.507063i	$-0.169269\pi$
$312$	0	0
$313$	−80.4580	−0.257054	−0.128527	−	0.991706i	$-0.541025\pi$
$313$	−80.4580	−0.257054	−0.128527	+	0.991706i	$0.541025\pi$
$314$	0	0
$315$	−22.6775	−0.0719921
$316$	0	0
$317$	− 135.504i	− 0.427458i	−0.976893	−	0.213729i	$-0.931439\pi$
$317$	− 135.504i	− 0.427458i	0.976893	−	0.213729i	$-0.0685609\pi$
$318$	0	0
$319$	− 463.070i	− 1.45163i
$320$	0	0
$321$	−257.001	−0.800626
$322$	0	0
$323$	−657.249	−2.03483
$324$	0	0
$325$	− 94.0163i	− 0.289281i
$326$	0	0
$327$	292.710i	0.895139i
$328$	0	0
$329$	102.658	0.312030
$330$	0	0
$331$	−435.738	−1.31643	−0.658214	−	0.752831i	$-0.728688\pi$
$331$	−435.738	−1.31643	−0.658214	+	0.752831i	$0.728688\pi$
$332$	0	0
$333$	259.013i	0.777818i
$334$	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
$339$	−35.8670	−0.105802
$340$	0	0
$341$	187.461i	0.549738i
$342$	0	0
$343$	− 246.964i	− 0.720012i
$344$	0	0
$345$	−45.1278	−0.130805
$346$	0	0
$347$	−232.394	−0.669723	−0.334862	−	0.942267i	$-0.608690\pi$
$347$	−232.394	−0.669723	−0.334862	+	0.942267i	$0.608690\pi$
$348$	0	0
$349$	458.147i	1.31274i	0.754438	+	0.656371i	$0.227909\pi$
$349$	458.147i	1.31274i	−0.754438	+	0.656371i	$0.772091\pi$
$350$	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
$355$	190.570	0.536816
$356$	0	0
$357$	− 147.743i	− 0.413845i
$358$	0	0
$359$	− 193.553i	− 0.539145i	−0.962980	−	0.269572i	$-0.913118\pi$
$359$	− 193.553i	− 0.539145i	0.962980	−	0.269572i	$-0.0868824\pi$
$360$	0	0
$361$	416.181	1.15286
$362$	0	0
$363$	−581.961	−1.60320
$364$	0	0
$365$	− 33.8224i	− 0.0926641i
$366$	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
$371$	116.531	0.314100
$372$	0	0
$373$	− 721.093i	− 1.93322i	−0.256244	−	0.966612i	$-0.582485\pi$
$373$	− 721.093i	− 1.93322i	0.256244	−	0.966612i	$-0.417515\pi$
$374$	0	0
$375$	25.6928i	0.0685142i
$376$	0	0
$377$	−450.094	−1.19388
$378$	0	0
$379$	−323.886	−0.854580	−0.427290	−	0.904115i	$-0.640532\pi$
$379$	−323.886	−0.854580	−0.427290	+	0.904115i	$0.640532\pi$
$380$	0	0
$381$	340.308i	0.893197i
$382$	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
$387$	104.913	0.271093
$388$	0	0
$389$	− 353.367i	− 0.908398i	−0.890900	−	0.454199i	$-0.849925\pi$
$389$	− 353.367i	− 0.908398i	0.890900	−	0.454199i	$-0.150075\pi$
$390$	0	0
$391$	207.048i	0.529535i
$392$	0	0
$393$	−397.143	−1.01054
$394$	0	0
$395$	289.111	0.731926
$396$	0	0
$397$	− 32.9297i	− 0.0829463i	−0.999140	−	0.0414732i	$-0.986795\pi$
$397$	− 32.9297i	− 0.0829463i	0.999140	−	0.0414732i	$-0.0132051\pi$
$398$	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
$403$	182.208	0.452128
$404$	0	0
$405$	− 75.3498i	− 0.186049i
$406$	0	0
$407$	1347.31i	3.31035i
$408$	0	0
$409$	114.925	0.280990	0.140495	−	0.990081i	$-0.455131\pi$
$409$	114.925	0.280990	0.140495	+	0.990081i	$0.455131\pi$
$410$	0	0
$411$	−373.132	−0.907864
$412$	0	0
$413$	− 50.0043i	− 0.121076i
$414$	0	0
$415$	100.100i	0.241204i
$416$	0	0
$417$	172.655	0.414041
$418$	0	0
$419$	−526.123	−1.25566	−0.627831	−	0.778349i	$-0.716057\pi$
$419$	−526.123	−1.25566	−0.627831	+	0.778349i	$0.716057\pi$
$420$	0	0
$421$	298.264i	0.708465i	0.935157	+	0.354233i	$0.115258\pi$
$421$	298.264i	0.708465i	−0.935157	+	0.354233i	$0.884742\pi$
$422$	0	0
$423$	− 140.005i	− 0.330980i
$424$	0	0
$425$	117.880	0.277364
$426$	0	0
$427$	96.1065	0.225074
$428$	0	0
$429$	835.924i	1.94854i
$430$	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
$435$	123.002	0.282763
$436$	0	0
$437$	− 244.829i	− 0.560250i
$438$	0	0
$439$	− 716.216i	− 1.63147i	−0.578425	−	0.815735i	$-0.696333\pi$
$439$	− 716.216i	− 1.63147i	0.578425	−	0.815735i	$-0.303667\pi$
$440$	0	0
$441$	−154.577	−0.350514
$442$	0	0
$443$	−653.956	−1.47620	−0.738100	−	0.674692i	$-0.764277\pi$
$443$	−653.956	−1.47620	−0.738100	+	0.674692i	$0.764277\pi$
$444$	0	0
$445$	− 181.975i	− 0.408933i
$446$	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
$451$	633.127	1.40383
$452$	0	0
$453$	279.817i	0.617698i
$454$	0	0
$455$	114.656i	0.251992i
$456$	0	0
$457$	−347.650	−0.760721	−0.380361	−	0.924838i	$-0.624200\pi$
$457$	−347.650	−0.760721	−0.380361	+	0.924838i	$0.624200\pi$
$458$	0	0
$459$	−689.096	−1.50130
$460$	0	0
$461$	− 329.633i	− 0.715039i	−0.933906	−	0.357520i	$-0.883623\pi$
$461$	− 329.633i	− 0.715039i	0.933906	−	0.357520i	$-0.116377\pi$
$462$	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
$467$	−595.434	−1.27502	−0.637510	−	0.770442i	$-0.720035\pi$
$467$	−595.434	−1.27502	−0.637510	+	0.770442i	$0.720035\pi$
$468$	0	0
$469$	− 128.383i	− 0.273738i
$470$	0	0
$471$	− 224.626i	− 0.476914i
$472$	0	0
$473$	545.727	1.15376
$474$	0	0
$475$	−139.390	−0.293452
$476$	0	0
$477$	− 158.925i	− 0.333176i
$478$	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
$483$	55.0350	0.113944
$484$	0	0
$485$	251.614i	0.518792i
$486$	0	0
$487$	− 692.389i	− 1.42174i	−0.703321	−	0.710872i	$-0.748300\pi$
$487$	− 692.389i	− 1.42174i	0.703321	−	0.710872i	$-0.251700\pi$
$488$	0	0
$489$	−211.607	−0.432735
$490$	0	0
$491$	253.027	0.515330	0.257665	−	0.966234i	$-0.417047\pi$
$491$	253.027	0.515330	0.257665	+	0.966234i	$0.417047\pi$
$492$	0	0
$493$	− 564.337i	− 1.14470i
$494$	0	0
$495$	160.876i	0.325003i
$496$	0	0
$497$	−232.407	−0.467619
$498$	0	0
$499$	−856.886	−1.71721	−0.858604	−	0.512640i	$-0.828668\pi$
$499$	−856.886	−1.71721	−0.858604	+	0.512640i	$0.828668\pi$
$500$	0	0
$501$	− 346.531i	− 0.691679i
$502$	0	0
$503$	− 53.5054i	− 0.106373i	−0.998585	−	0.0531863i	$-0.983062\pi$
$503$	− 53.5054i	− 0.106373i	0.998585	−	0.0531863i	$-0.0169377\pi$
$504$	0	0
$505$	−142.241	−0.281665
$506$	0	0
$507$	424.132	0.836552
$508$	0	0
$509$	− 313.259i	− 0.615439i	−0.951477	−	0.307720i	$-0.900434\pi$
$509$	− 313.259i	− 0.615439i	0.951477	−	0.307720i	$-0.0995659\pi$
$510$	0	0
$511$	41.2477i	0.0807195i
$512$	0	0
$513$	814.839	1.58838
$514$	0	0
$515$	170.566	0.331196
$516$	0	0
$517$	− 728.264i	− 1.40864i
$518$	0	0
$519$	87.1468i	0.167913i
$520$	0	0
$521$	−291.820	−0.560116	−0.280058	−	0.959983i	$-0.590354\pi$
$521$	−291.820	−0.560116	−0.280058	+	0.959983i	$0.590354\pi$
$522$	0	0
$523$	236.297	0.451811	0.225906	−	0.974149i	$-0.427466\pi$
$523$	236.297	0.451811	0.225906	+	0.974149i	$0.427466\pi$
$524$	0	0
$525$	− 31.3334i	− 0.0596826i
$526$	0	0
$527$	228.456i	0.433503i
$528$	0	0
$529$	451.873	0.854203
$530$	0	0
$531$	−68.1958	−0.128429
$532$	0	0
$533$	− 615.386i	− 1.15457i
$534$	0	0
$535$	250.071i	0.467422i
$536$	0	0
$537$	819.934	1.52688
$538$	0	0
$539$	−804.064	−1.49177
$540$	0	0
$541$	− 479.488i	− 0.886300i	−0.896448	−	0.443150i	$-0.853861\pi$
$541$	− 479.488i	− 0.886300i	0.896448	−	0.443150i	$-0.146139\pi$
$542$	0	0
$543$	− 760.484i	− 1.40052i
$544$	0	0
$545$	284.817	0.522601
$546$	0	0
$547$	606.330	1.10846	0.554232	−	0.832362i	$-0.313012\pi$
$547$	606.330	1.10846	0.554232	+	0.832362i	$0.313012\pi$
$548$	0	0
$549$	− 131.070i	− 0.238743i
$550$	0	0
$551$	667.315i	1.21110i
$552$	0	0
$553$	−352.582	−0.637580
$554$	0	0
$555$	−357.877	−0.644823
$556$	0	0
$557$	22.2604i	0.0399649i	0.999800	+	0.0199824i	$0.00636103\pi$
$557$	22.2604i	0.0399649i	−0.999800	+	0.0199824i	$0.993639\pi$
$558$	0	0
$559$	− 530.435i	− 0.948899i
$560$	0	0
$561$	−1048.10	−1.86827
$562$	0	0
$563$	−518.786	−0.921468	−0.460734	−	0.887538i	$-0.652414\pi$
$563$	−518.786	−0.921468	−0.460734	+	0.887538i	$0.652414\pi$
$564$	0	0
$565$	34.8998i	0.0617696i
$566$	0	0
$567$	91.8919i	0.162067i
$568$	0	0
$569$	596.499	1.04833	0.524164	−	0.851617i	$-0.324378\pi$
$569$	596.499	1.04833	0.524164	+	0.851617i	$0.324378\pi$
$570$	0	0
$571$	−290.657	−0.509031	−0.254515	−	0.967069i	$-0.581916\pi$
$571$	−290.657	−0.509031	−0.254515	+	0.967069i	$0.581916\pi$
$572$	0	0
$573$	823.438i	1.43706i
$574$	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
$579$	−202.928	−0.350480
$580$	0	0
$581$	− 122.075i	− 0.210112i
$582$	0	0
$583$	− 826.683i	− 1.41798i
$584$	0	0
$585$	156.368	0.267296
$586$	0	0
$587$	1147.72	1.95522	0.977611	−	0.210418i	$-0.0674826\pi$
$587$	1147.72	1.95522	0.977611	+	0.210418i	$0.0674826\pi$
$588$	0	0
$589$	− 270.143i	− 0.458648i
$590$	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
$595$	−143.759	−0.241611
$596$	0	0
$597$	274.879i	0.460433i
$598$	0	0
$599$	602.645i	1.00609i	0.864262	+	0.503043i	$0.167786\pi$
$599$	602.645i	1.00609i	−0.864262	+	0.503043i	$0.832214\pi$
$600$	0	0
$601$	−804.359	−1.33837	−0.669184	−	0.743097i	$-0.733356\pi$
$601$	−804.359	−1.33837	−0.669184	+	0.743097i	$0.733356\pi$
$602$	0	0
$603$	−175.088	−0.290362
$604$	0	0
$605$	566.269i	0.935981i
$606$	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
$611$	−707.857	−1.15852
$612$	0	0
$613$	369.141i	0.602188i	0.953595	+	0.301094i	$0.0973517\pi$
$613$	369.141i	0.602188i	−0.953595	+	0.301094i	$0.902648\pi$
$614$	0	0
$615$	168.173i	0.273452i
$616$	0	0
$617$	31.3293	0.0507768	0.0253884	−	0.999678i	$-0.491918\pi$
$617$	31.3293	0.0507768	0.0253884	+	0.999678i	$0.491918\pi$
$618$	0	0
$619$	472.149	0.762761	0.381380	−	0.924418i	$-0.375449\pi$
$619$	472.149	0.762761	0.381380	+	0.924418i	$0.375449\pi$
$620$	0	0
$621$	− 256.693i	− 0.413353i
$622$	0	0
$623$	221.925i	0.356221i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	1239.35	1.97664
$628$	0	0
$629$	1641.95i	2.61042i
$630$	0	0
$631$	− 1107.24i	− 1.75474i	−0.479815	−	0.877369i	$-0.659296\pi$
$631$	− 1107.24i	− 1.75474i	0.479815	−	0.877369i	$-0.340704\pi$
$632$	0	0
$633$	−196.518	−0.310455
$634$	0	0
$635$	331.132	0.521467
$636$	0	0
$637$	781.532i	1.22690i
$638$	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
$643$	−151.870	−0.236190	−0.118095	−	0.993002i	$-0.537679\pi$
$643$	−151.870	−0.236190	−0.118095	+	0.993002i	$0.537679\pi$
$644$	0	0
$645$	144.957i	0.224740i
$646$	0	0
$647$	590.497i	0.912670i	0.889808	+	0.456335i	$0.150838\pi$
$647$	590.497i	0.912670i	−0.889808	+	0.456335i	$0.849162\pi$
$648$	0	0
$649$	−354.735	−0.546587
$650$	0	0
$651$	60.7254	0.0932802
$652$	0	0
$653$	− 356.873i	− 0.546514i	−0.961941	−	0.273257i	$-0.911899\pi$
$653$	− 356.873i	− 0.546514i	0.961941	−	0.273257i	$-0.0881009\pi$
$654$	0	0
$655$	386.434i	0.589975i
$656$	0	0
$657$	56.2535	0.0856218
$658$	0	0
$659$	309.347	0.469419	0.234709	−	0.972066i	$-0.424586\pi$
$659$	309.347	0.469419	0.234709	+	0.972066i	$0.424586\pi$
$660$	0	0
$661$	432.538i	0.654369i	0.944960	+	0.327185i	$0.106100\pi$
$661$	432.538i	0.654369i	−0.944960	+	0.327185i	$0.893900\pi$
$662$	0	0
$663$	1018.73i	1.53655i
$664$	0	0
$665$	169.991	0.255626
$666$	0	0
$667$	210.219	0.315171
$668$	0	0
$669$	− 634.347i	− 0.948202i
$670$	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
$675$	−146.144	−0.216510
$676$	0	0
$677$	596.363i	0.880890i	0.897780	+	0.440445i	$0.145179\pi$
$677$	596.363i	0.880890i	−0.897780	+	0.440445i	$0.854821\pi$
$678$	0	0
$679$	− 306.853i	− 0.451919i
$680$	0	0
$681$	200.577	0.294533
$682$	0	0
$683$	845.172	1.23744	0.618720	−	0.785612i	$-0.287652\pi$
$683$	845.172	1.23744	0.618720	+	0.785612i	$0.287652\pi$
$684$	0	0
$685$	363.070i	0.530030i
$686$	0	0
$687$	227.789i	0.331570i
$688$	0	0
$689$	−803.518	−1.16621
$690$	0	0
$691$	99.7834	0.144404	0.0722022	−	0.997390i	$-0.476997\pi$
$691$	99.7834	0.144404	0.0722022	+	0.997390i	$0.476997\pi$
$692$	0	0
$693$	− 196.195i	− 0.283109i
$694$	0	0
$695$	− 167.999i	− 0.241726i
$696$	0	0
$697$	771.584	1.10701
$698$	0	0
$699$	−380.214	−0.543940
$700$	0	0
$701$	258.057i	0.368127i	0.982914	+	0.184063i	$0.0589252\pi$
$701$	258.057i	0.368127i	−0.982914	+	0.184063i	$0.941075\pi$
$702$	0	0
$703$	− 1941.57i	− 2.76183i
$704$	0	0
$705$	193.443	0.274388
$706$	0	0
$707$	173.468	0.245358
$708$	0	0
$709$	− 463.455i	− 0.653674i	−0.945081	−	0.326837i	$-0.894017\pi$
$709$	− 463.455i	− 0.653674i	0.945081	−	0.326837i	$-0.105983\pi$
$710$	0	0
$711$	480.850i	0.676301i
$712$	0	0
$713$	−85.1012	−0.119357
$714$	0	0
$715$	813.383	1.13760
$716$	0	0
$717$	173.107i	0.241432i
$718$	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
$723$	583.113	0.806519
$724$	0	0
$725$	− 119.685i	− 0.165083i
$726$	0	0
$727$	450.184i	0.619236i	0.950861	+	0.309618i	$0.100201\pi$
$727$	450.184i	0.619236i	−0.950861	+	0.309618i	$0.899799\pi$
$728$	0	0
$729$	729.841	1.00115
$730$	0	0
$731$	665.070	0.909808
$732$	0	0
$733$	− 1413.12i	− 1.92786i	−0.266160	−	0.963929i	$-0.585755\pi$
$733$	− 1413.12i	− 1.92786i	0.266160	−	0.963929i	$-0.414245\pi$
$734$	0	0
$735$	− 213.577i	− 0.290581i
$736$	0	0
$737$	−910.760	−1.23577
$738$	0	0
$739$	331.195	0.448167	0.224083	−	0.974570i	$-0.428061\pi$
$739$	331.195	0.448167	0.224083	+	0.974570i	$0.428061\pi$
$740$	0	0
$741$	− 1204.62i	− 1.62567i
$742$	0	0
$743$	− 168.111i	− 0.226259i	−0.993580	−	0.113130i	$-0.963912\pi$
$743$	− 168.111i	− 0.226259i	0.993580	−	0.113130i	$-0.0360875\pi$
$744$	0	0
$745$	573.464	0.769750
$746$	0	0
$747$	−166.486	−0.222873
$748$	0	0
$749$	− 304.971i	− 0.407171i
$750$	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
$755$	272.272	0.360625
$756$	0	0
$757$	1184.73i	1.56503i	0.622633	+	0.782514i	$0.286063\pi$
$757$	1184.73i	1.56503i	−0.622633	+	0.782514i	$0.713937\pi$
$758$	0	0
$759$	− 390.423i	− 0.514392i
$760$	0	0
$761$	136.691	0.179620	0.0898099	−	0.995959i	$-0.471374\pi$
$761$	136.691	0.179620	0.0898099	+	0.995959i	$0.471374\pi$
$762$	0	0
$763$	−347.346	−0.455237
$764$	0	0
$765$	196.058i	0.256285i
$766$	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
$771$	−946.975	−1.22824
$772$	0	0
$773$	209.531i	0.271063i	0.990773	+	0.135531i	$0.0432741\pi$
$773$	209.531i	0.271063i	−0.990773	+	0.135531i	$0.956726\pi$
$774$	0	0
$775$	48.4511i	0.0625175i
$776$	0	0
$777$	436.444	0.561704
$778$	0	0
$779$	−912.379	−1.17122
$780$	0	0
$781$	1648.71i	2.11103i
$782$	0	0
$783$	699.650i	0.893550i
$784$	0	0
$785$	−218.569	−0.278432
$786$	0	0
$787$	176.067	0.223719	0.111859	−	0.993724i	$-0.464319\pi$
$787$	176.067	0.223719	0.111859	+	0.993724i	$0.464319\pi$
$788$	0	0
$789$	− 780.365i	− 0.989056i
$790$	0	0
$791$	− 42.5617i	− 0.0538074i
$792$	0	0
$793$	−662.683	−0.835666
$794$	0	0
$795$	219.586	0.276208
$796$	0	0
$797$	− 1185.07i	− 1.48692i	−0.668783	−	0.743458i	$-0.733184\pi$
$797$	− 1185.07i	− 1.48692i	0.668783	−	0.743458i	$-0.266816\pi$
$798$	0	0
$799$	− 887.526i	− 1.11080i
$800$	0	0
$801$	302.662	0.377855
$802$	0	0
$803$	292.615	0.364402
$804$	0	0
$805$	− 53.5510i	− 0.0665230i
$806$	0	0
$807$	275.107i	0.340901i
$808$	0	0
$809$	−537.511	−0.664414	−0.332207	−	0.943206i	$-0.607793\pi$
$809$	−537.511	−0.664414	−0.332207	+	0.943206i	$0.607793\pi$
$810$	0	0
$811$	859.330	1.05959	0.529797	−	0.848125i	$-0.322268\pi$
$811$	859.330	1.05959	0.529797	+	0.848125i	$0.322268\pi$
$812$	0	0
$813$	− 930.332i	− 1.14432i
$814$	0	0
$815$	205.901i	0.252640i
$816$	0	0
$817$	−786.429	−0.962581
$818$	0	0
$819$	−190.697	−0.232841
$820$	0	0
$821$	808.044i	0.984220i	0.870533	+	0.492110i	$0.163774\pi$
$821$	808.044i	0.984220i	−0.870533	+	0.492110i	$0.836226\pi$
$822$	0	0
$823$	− 709.287i	− 0.861831i	−0.902392	−	0.430916i	$-0.858191\pi$
$823$	− 709.287i	− 0.861831i	0.902392	−	0.430916i	$-0.141809\pi$
$824$	0	0
$825$	−222.282	−0.269432
$826$	0	0
$827$	714.799	0.864328	0.432164	−	0.901795i	$-0.357750\pi$
$827$	714.799	0.864328	0.432164	+	0.901795i	$0.357750\pi$
$828$	0	0
$829$	730.675i	0.881393i	0.897656	+	0.440697i	$0.145269\pi$
$829$	730.675i	0.881393i	−0.897656	+	0.440697i	$0.854731\pi$
$830$	0	0
$831$	57.8059i	0.0695618i
$832$	0	0
$833$	−979.902	−1.17635
$834$	0	0
$835$	−337.187	−0.403817
$836$	0	0
$837$	− 283.233i	− 0.338391i
$838$	0	0
$839$	− 1154.02i	− 1.37548i	−0.725959	−	0.687738i	$-0.758604\pi$
$839$	− 1154.02i	− 1.37548i	0.725959	−	0.687738i	$-0.241396\pi$
$840$	0	0
$841$	268.019	0.318691
$842$	0	0
$843$	−1044.45	−1.23897
$844$	0	0
$845$	− 412.695i	− 0.488397i
$846$	0	0
$847$	− 690.586i	− 0.815332i
$848$	0	0
$849$	−256.182	−0.301746
$850$	0	0
$851$	−611.638	−0.718728
$852$	0	0
$853$	− 105.220i	− 0.123353i	−0.998096	−	0.0616764i	$-0.980355\pi$
$853$	− 105.220i	− 0.123353i	0.998096	−	0.0616764i	$-0.0196447\pi$
$854$	0	0
$855$	− 231.834i	− 0.271150i
$856$	0	0
$857$	681.033	0.794671	0.397336	−	0.917673i	$-0.369935\pi$
$857$	681.033	0.794671	0.397336	+	0.917673i	$0.369935\pi$
$858$	0	0
$859$	921.542	1.07281	0.536404	−	0.843961i	$-0.319782\pi$
$859$	921.542	1.07281	0.536404	+	0.843961i	$0.319782\pi$
$860$	0	0
$861$	− 205.093i	− 0.238203i
$862$	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
$867$	−613.172	−0.707235
$868$	0	0
$869$	2501.25i	2.87830i
$870$	0	0
$871$	885.238i	1.01635i
$872$	0	0
$873$	−418.486	−0.479365
$874$	0	0
$875$	−30.4884	−0.0348439
$876$	0	0
$877$	− 795.922i	− 0.907551i	−0.891116	−	0.453775i	$-0.850077\pi$
$877$	− 795.922i	− 0.907551i	0.891116	−	0.453775i	$-0.149923\pi$
$878$	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
$883$	−217.973	−0.246855	−0.123427	−	0.992354i	$-0.539389\pi$
$883$	−217.973	−0.246855	−0.123427	+	0.992354i	$0.539389\pi$
$884$	0	0
$885$	− 94.2255i	− 0.106470i
$886$	0	0
$887$	− 1236.92i	− 1.39450i	−0.716829	−	0.697249i	$-0.754407\pi$
$887$	− 1236.92i	− 1.39450i	0.716829	−	0.697249i	$-0.245593\pi$
$888$	0	0
$889$	−403.827	−0.454249
$890$	0	0
$891$	651.889	0.731638
$892$	0	0
$893$	1049.48i	1.17523i
$894$	0	0
$895$	− 797.824i	− 0.891424i
$896$	0	0
$897$	−379.483	−0.423058
$898$	0	0
$899$	231.955	0.258014
$900$	0	0
$901$	− 1007.47i	− 1.11817i
$902$	0	0
$903$	− 176.781i	− 0.195771i
$904$	0	0
$905$	−739.978	−0.817655
$906$	0	0
$907$	804.548	0.887043	0.443522	−	0.896264i	$-0.353729\pi$
$907$	804.548	0.887043	0.443522	+	0.896264i	$0.353729\pi$
$908$	0	0
$909$	− 236.575i	− 0.260259i
$910$	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
$915$	181.098	0.197921
$916$	0	0
$917$	− 471.270i	− 0.513926i
$918$	0	0
$919$	− 502.728i	− 0.547039i	−0.961867	−	0.273519i	$-0.911812\pi$
$919$	− 502.728i	− 0.547039i	0.961867	−	0.273519i	$-0.0881877\pi$
$920$	0	0
$921$	−1173.23	−1.27386
$922$	0	0
$923$	1602.51	1.73620
$924$	0	0
$925$	348.226i	0.376461i
$926$	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
$931$	1158.71	1.24459
$932$	0	0
$933$	724.784i	0.776832i
$934$	0	0
$935$	1019.84i	1.09073i
$936$	0	0
$937$	−93.6571	−0.0999542	−0.0499771	−	0.998750i	$-0.515915\pi$
$937$	−93.6571	−0.0999542	−0.0499771	+	0.998750i	$0.515915\pi$
$938$	0	0
$939$	184.895	0.196907
$940$	0	0
$941$	93.5311i	0.0993954i	0.998764	+	0.0496977i	$0.0158258\pi$
$941$	93.5311i	0.0993954i	−0.998764	+	0.0496977i	$0.984174\pi$
$942$	0	0
$943$	287.420i	0.304793i
$944$	0	0
$945$	178.228	0.188601
$946$	0	0
$947$	1061.81	1.12124	0.560619	−	0.828074i	$-0.310563\pi$
$947$	1061.81	1.12124	0.560619	+	0.828074i	$0.310563\pi$
$948$	0	0
$949$	− 284.415i	− 0.299700i
$950$	0	0
$951$	311.393i	0.327438i
$952$	0	0
$953$	435.168	0.456630	0.228315	−	0.973587i	$-0.426678\pi$
$953$	435.168	0.456630	0.228315	+	0.973587i	$0.426678\pi$
$954$	0	0
$955$	801.234	0.838988
$956$	0	0
$957$	1064.15i	1.11197i
$958$	0	0
$959$	− 442.778i	− 0.461708i
$960$	0	0
$961$	867.100	0.902289
$962$	0	0
$963$	−415.919	−0.431899
$964$	0	0
$965$	197.456i	0.204618i
$966$	0	0
$967$	− 1400.02i	− 1.44780i	−0.689905	−	0.723900i	$-0.742348\pi$
$967$	− 1400.02i	− 1.44780i	0.689905	−	0.723900i	$-0.257652\pi$
$968$	0	0
$969$	1510.38	1.55870
$970$	0	0
$971$	−562.184	−0.578974	−0.289487	−	0.957182i	$-0.593485\pi$
$971$	−562.184	−0.578974	−0.289487	+	0.957182i	$0.593485\pi$
$972$	0	0
$973$	204.881i	0.210567i
$974$	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
$979$	1574.36	1.60813
$980$	0	0
$981$	473.709i	0.482884i
$982$	0	0
$983$	239.801i	0.243948i	0.992533	+	0.121974i	$0.0389224\pi$
$983$	239.801i	0.243948i	−0.992533	+	0.121974i	$0.961078\pi$
$984$	0	0
$985$	−212.166	−0.215397
$986$	0	0
$987$	−235.911	−0.239019
$988$	0	0
$989$	247.743i	0.250498i
$990$	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
$995$	267.466	0.268810
$996$	0	0
$997$	127.529i	0.127913i	0.997953	+	0.0639565i	$0.0203719\pi$
$997$	127.529i	0.127913i	−0.997953	+	0.0639565i	$0.979628\pi$
$998$	0	0
$999$	− 2035.65i	− 2.03769i

Twists

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

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

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.29803 q^{3} +2.23607i q^{5} -2.72697i q^{7} -3.71904 q^{9} +O(q^{10})\)	\(q-2.29803 q^{3} +2.23607i q^{5} -2.72697i q^{7} -3.71904 q^{9} -19.3454 q^{11} +18.8033i q^{13} -5.13856i q^{15} -23.5759 q^{17} +27.8780 q^{19} +6.26667i q^{21} -8.78218i q^{23} -5.00000 q^{25} +29.2288 q^{27} +23.9370i q^{29} -9.69022i q^{31} +44.4563 q^{33} +6.09769 q^{35} -69.6453i q^{37} -43.2106i q^{39} -32.7276 q^{41} -28.2097 q^{43} -8.31602i q^{45} +37.6454i q^{47} +41.5636 q^{49} +54.1783 q^{51} +42.7329i q^{53} -43.2575i q^{55} -64.0645 q^{57} +18.3369 q^{59} +35.2430i q^{61} +10.1417i q^{63} -42.0454 q^{65} +47.0790 q^{67} +20.1818i q^{69} -85.2253i q^{71} -15.1258 q^{73} +11.4902 q^{75} +52.7542i q^{77} -129.294i q^{79} -33.6974 q^{81} +44.7659 q^{83} -52.7174i q^{85} -55.0081i q^{87} -81.3817 q^{89} +51.2759 q^{91} +22.2685i q^{93} +62.3370i q^{95} +112.525 q^{97} +71.9461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 48 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 48 * q^9	\( 16 q - 16 q^{3} + 48 q^{9} - 64 q^{11} - 96 q^{19} - 80 q^{25} - 64 q^{27} + 32 q^{33} + 80 q^{35} + 96 q^{41} + 176 q^{43} - 176 q^{49} - 96 q^{51} - 352 q^{57} + 32 q^{59} + 560 q^{67} + 320 q^{73} + 80 q^{75} - 48 q^{81} - 48 q^{83} + 96 q^{89} + 32 q^{91} + 448 q^{97} - 704 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 48 * q^9 - 64 * q^11 - 96 * q^19 - 80 * q^25 - 64 * q^27 + 32 * q^33 + 80 * q^35 + 96 * q^41 + 176 * q^43 - 176 * q^49 - 96 * q^51 - 352 * q^57 + 32 * q^59 + 560 * q^67 + 320 * q^73 + 80 * q^75 - 48 * q^81 - 48 * q^83 + 96 * q^89 + 32 * q^91 + 448 * q^97 - 704 * q^99

Introduction

L-functions

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Database

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