LMFDB - Embedded newform 1280.3.g.c.1151.2

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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1280.1151
Dual form			1280.3.g.c.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{3} +2.23607i q^{5} +1.23607i q^{7} -7.47214 q^{9} +11.4164 q^{11} -5.41641i q^{13} -2.76393i q^{15} +6.94427 q^{17} -29.8885 q^{19} -1.52786i q^{21} -19.1246i q^{23} -5.00000 q^{25} +20.3607 q^{27} -21.0557i q^{29} +34.4721i q^{31} -14.1115 q^{33} -2.76393 q^{35} -19.3050i q^{37} +6.69505i q^{39} +58.1378 q^{41} -62.7639 q^{43} -16.7082i q^{45} +63.4853i q^{47} +47.4721 q^{49} -8.58359 q^{51} -98.1378i q^{53} +25.5279i q^{55} +36.9443 q^{57} +19.2786 q^{59} -1.19350i q^{61} -9.23607i q^{63} +12.1115 q^{65} +5.01316 q^{67} +23.6393i q^{69} -84.3607i q^{71} +70.7214 q^{73} +6.18034 q^{75} +14.1115i q^{77} -124.498i q^{79} +42.0820 q^{81} +160.652 q^{83} +15.5279i q^{85} +26.0263i q^{87} -46.2229 q^{89} +6.69505 q^{91} -42.6099i q^{93} -66.8328i q^{95} +133.331 q^{97} -85.3050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 12 q^{9} - 8 q^{11} - 8 q^{17} - 48 q^{19} - 20 q^{25} - 8 q^{27} - 128 q^{33} - 20 q^{35} - 260 q^{43} + 172 q^{49} - 88 q^{51} + 112 q^{57} + 256 q^{59} + 120 q^{65} - 132 q^{67} + 104 q^{73}+ \cdots - 216 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 12 * q^9 - 8 * q^11 - 8 * q^17 - 48 * q^19 - 20 * q^25 - 8 * q^27 - 128 * q^33 - 20 * q^35 - 260 * q^43 + 172 * q^49 - 88 * q^51 + 112 * q^57 + 256 * q^59 + 120 * q^65 - 132 * q^67 + 104 * q^73 - 20 * q^75 - 100 * q^81 + 580 * q^83 - 328 * q^89 + 152 * q^91 + 104 * q^97 - 216 * q^99

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
$3$	−1.23607	−0.412023	−0.206011	−	0.978550i	$-0.566048\pi$
$3$	−1.23607	−0.412023	−0.206011	+	0.978550i	$0.566048\pi$
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	1.23607i	0.176581i	0.996095	+	0.0882906i	$0.0281404\pi$
$7$	1.23607i	0.176581i	−0.996095	+	0.0882906i	$0.971860\pi$
$8$	0	0
$9$	−7.47214	−0.830237
$10$	0	0
$11$	11.4164	1.03786	0.518928	−	0.854818i	$-0.326331\pi$
$11$	11.4164	1.03786	0.518928	+	0.854818i	$0.326331\pi$
$12$	0	0
$13$	− 5.41641i	− 0.416647i	−0.978060	−	0.208323i	$-0.933199\pi$
$13$	− 5.41641i	− 0.416647i	0.978060	−	0.208323i	$-0.0668006\pi$
$14$	0	0
$15$	− 2.76393i	− 0.184262i
$16$	0	0
$17$	6.94427	0.408487	0.204243	−	0.978920i	$-0.434527\pi$
$17$	6.94427	0.408487	0.204243	+	0.978920i	$0.434527\pi$
$18$	0	0
$19$	−29.8885	−1.57308	−0.786541	−	0.617539i	$-0.788130\pi$
$19$	−29.8885	−1.57308	−0.786541	+	0.617539i	$0.788130\pi$
$20$	0	0
$21$	− 1.52786i	− 0.0727554i
$22$	0	0
$23$	− 19.1246i	− 0.831505i	−0.909478	−	0.415752i	$-0.863518\pi$
$23$	− 19.1246i	− 0.831505i	0.909478	−	0.415752i	$-0.136482\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	20.3607	0.754099
$28$	0	0
$29$	− 21.0557i	− 0.726060i	−0.931778	−	0.363030i	$-0.881742\pi$
$29$	− 21.0557i	− 0.726060i	0.931778	−	0.363030i	$-0.118258\pi$
$30$	0	0
$31$	34.4721i	1.11200i	0.831181	+	0.556002i	$0.187665\pi$
$31$	34.4721i	1.11200i	−0.831181	+	0.556002i	$0.812335\pi$
$32$	0	0
$33$	−14.1115	−0.427620
$34$	0	0
$35$	−2.76393	−0.0789695
$36$	0	0
$37$	− 19.3050i	− 0.521755i	−0.965372	−	0.260878i	$-0.915988\pi$
$37$	− 19.3050i	− 0.521755i	0.965372	−	0.260878i	$-0.0840119\pi$
$38$	0	0
$39$	6.69505i	0.171668i
$40$	0	0
$41$	58.1378	1.41799	0.708997	−	0.705211i	$-0.249148\pi$
$41$	58.1378	1.41799	0.708997	+	0.705211i	$0.249148\pi$
$42$	0	0
$43$	−62.7639	−1.45963	−0.729813	−	0.683647i	$-0.760393\pi$
$43$	−62.7639	−1.45963	−0.729813	+	0.683647i	$0.760393\pi$
$44$	0	0
$45$	− 16.7082i	− 0.371293i
$46$	0	0
$47$	63.4853i	1.35075i	0.737474	+	0.675375i	$0.236018\pi$
$47$	63.4853i	1.35075i	−0.737474	+	0.675375i	$0.763982\pi$
$48$	0	0
$49$	47.4721	0.968819
$50$	0	0
$51$	−8.58359	−0.168306
$52$	0	0
$53$	− 98.1378i	− 1.85166i	−0.377945	−	0.925828i	$-0.623369\pi$
$53$	− 98.1378i	− 1.85166i	0.377945	−	0.925828i	$-0.376631\pi$
$54$	0	0
$55$	25.5279i	0.464143i
$56$	0	0
$57$	36.9443	0.648145
$58$	0	0
$59$	19.2786	0.326757	0.163378	−	0.986563i	$-0.447761\pi$
$59$	19.2786	0.326757	0.163378	+	0.986563i	$0.447761\pi$
$60$	0	0
$61$	− 1.19350i	− 0.0195655i	−0.999952	−	0.00978275i	$-0.996886\pi$
$61$	− 1.19350i	− 0.0195655i	0.999952	−	0.00978275i	$-0.00311400\pi$
$62$	0	0
$63$	− 9.23607i	− 0.146604i
$64$	0	0
$65$	12.1115	0.186330
$66$	0	0
$67$	5.01316	0.0748232	0.0374116	−	0.999300i	$-0.488089\pi$
$67$	5.01316	0.0748232	0.0374116	+	0.999300i	$0.488089\pi$
$68$	0	0
$69$	23.6393i	0.342599i
$70$	0	0
$71$	− 84.3607i	− 1.18818i	−0.804399	−	0.594089i	$-0.797513\pi$
$71$	− 84.3607i	− 1.18818i	0.804399	−	0.594089i	$-0.202487\pi$
$72$	0	0
$73$	70.7214	0.968786	0.484393	−	0.874851i	$-0.339041\pi$
$73$	70.7214	0.968786	0.484393	+	0.874851i	$0.339041\pi$
$74$	0	0
$75$	6.18034	0.0824045
$76$	0	0
$77$	14.1115i	0.183266i
$78$	0	0
$79$	− 124.498i	− 1.57593i	−0.615720	−	0.787965i	$-0.711135\pi$
$79$	− 124.498i	− 1.57593i	0.615720	−	0.787965i	$-0.288865\pi$
$80$	0	0
$81$	42.0820	0.519531
$82$	0	0
$83$	160.652	1.93557	0.967786	−	0.251774i	$-0.0810140\pi$
$83$	160.652	1.93557	0.967786	+	0.251774i	$0.0810140\pi$
$84$	0	0
$85$	15.5279i	0.182681i
$86$	0	0
$87$	26.0263i	0.299153i
$88$	0	0
$89$	−46.2229	−0.519359	−0.259679	−	0.965695i	$-0.583617\pi$
$89$	−46.2229	−0.519359	−0.259679	+	0.965695i	$0.583617\pi$
$90$	0	0
$91$	6.69505	0.0735720
$92$	0	0
$93$	− 42.6099i	− 0.458171i
$94$	0	0
$95$	− 66.8328i	− 0.703503i
$96$	0	0
$97$	133.331	1.37455	0.687275	−	0.726398i	$-0.258807\pi$
$97$	133.331	1.37455	0.687275	+	0.726398i	$0.258807\pi$
$98$	0	0
$99$	−85.3050	−0.861666
$100$	0	0
$101$	− 49.7771i	− 0.492842i	−0.969163	−	0.246421i	$-0.920745\pi$
$101$	− 49.7771i	− 0.492842i	0.969163	−	0.246421i	$-0.0792547\pi$
$102$	0	0
$103$	196.705i	1.90976i	0.296995	+	0.954879i	$0.404016\pi$
$103$	196.705i	1.90976i	−0.296995	+	0.954879i	$0.595984\pi$
$104$	0	0
$105$	3.41641	0.0325372
$106$	0	0
$107$	−14.9017	−0.139268	−0.0696341	−	0.997573i	$-0.522183\pi$
$107$	−14.9017	−0.139268	−0.0696341	+	0.997573i	$0.522183\pi$
$108$	0	0
$109$	− 174.859i	− 1.60421i	−0.597182	−	0.802106i	$-0.703713\pi$
$109$	− 174.859i	− 1.60421i	0.597182	−	0.802106i	$-0.296287\pi$
$110$	0	0
$111$	23.8622i	0.214975i
$112$	0	0
$113$	−33.0557	−0.292529	−0.146264	−	0.989246i	$-0.546725\pi$
$113$	−33.0557	−0.292529	−0.146264	+	0.989246i	$0.546725\pi$
$114$	0	0
$115$	42.7639	0.371860
$116$	0	0
$117$	40.4721i	0.345916i
$118$	0	0
$119$	8.58359i	0.0721310i
$120$	0	0
$121$	9.33437	0.0771435
$122$	0	0
$123$	−71.8622	−0.584246
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	− 93.7345i	− 0.738067i	−0.929416	−	0.369034i	$-0.879689\pi$
$127$	− 93.7345i	− 0.738067i	0.929416	−	0.369034i	$-0.120311\pi$
$128$	0	0
$129$	77.5805	0.601399
$130$	0	0
$131$	−80.8065	−0.616844	−0.308422	−	0.951250i	$-0.599801\pi$
$131$	−80.8065	−0.616844	−0.308422	+	0.951250i	$0.599801\pi$
$132$	0	0
$133$	− 36.9443i	− 0.277776i
$134$	0	0
$135$	45.5279i	0.337243i
$136$	0	0
$137$	192.164	1.40266	0.701329	−	0.712838i	$-0.252591\pi$
$137$	192.164	1.40266	0.701329	+	0.712838i	$0.252591\pi$
$138$	0	0
$139$	−6.60990	−0.0475533	−0.0237766	−	0.999717i	$-0.507569\pi$
$139$	−6.60990	−0.0475533	−0.0237766	+	0.999717i	$0.507569\pi$
$140$	0	0
$141$	− 78.4721i	− 0.556540i
$142$	0	0
$143$	− 61.8359i	− 0.432419i
$144$	0	0
$145$	47.0820	0.324704
$146$	0	0
$147$	−58.6788	−0.399175
$148$	0	0
$149$	− 199.803i	− 1.34096i	−0.741927	−	0.670481i	$-0.766088\pi$
$149$	− 199.803i	− 1.34096i	0.741927	−	0.670481i	$-0.233912\pi$
$150$	0	0
$151$	− 53.2523i	− 0.352664i	−0.984331	−	0.176332i	$-0.943577\pi$
$151$	− 53.2523i	− 0.352664i	0.984331	−	0.176332i	$-0.0564233\pi$
$152$	0	0
$153$	−51.8885	−0.339141
$154$	0	0
$155$	−77.0820	−0.497303
$156$	0	0
$157$	− 43.8034i	− 0.279003i	−0.990222	−	0.139501i	$-0.955450\pi$
$157$	− 43.8034i	− 0.279003i	0.990222	−	0.139501i	$-0.0445499\pi$
$158$	0	0
$159$	121.305i	0.762924i
$160$	0	0
$161$	23.6393	0.146828
$162$	0	0
$163$	188.705	1.15770	0.578850	−	0.815434i	$-0.303502\pi$
$163$	188.705	1.15770	0.578850	+	0.815434i	$0.303502\pi$
$164$	0	0
$165$	− 31.5542i	− 0.191237i
$166$	0	0
$167$	− 260.705i	− 1.56111i	−0.625088	−	0.780554i	$-0.714937\pi$
$167$	− 260.705i	− 1.56111i	0.625088	−	0.780554i	$-0.285063\pi$
$168$	0	0
$169$	139.663	0.826405
$170$	0	0
$171$	223.331	1.30603
$172$	0	0
$173$	− 28.6950i	− 0.165867i	−0.996555	−	0.0829337i	$-0.973571\pi$
$173$	− 28.6950i	− 0.165867i	0.996555	−	0.0829337i	$-0.0264290\pi$
$174$	0	0
$175$	− 6.18034i	− 0.0353162i
$176$	0	0
$177$	−23.8297	−0.134631
$178$	0	0
$179$	95.5016	0.533528	0.266764	−	0.963762i	$-0.414046\pi$
$179$	95.5016	0.533528	0.266764	+	0.963762i	$0.414046\pi$
$180$	0	0
$181$	− 176.885i	− 0.977268i	−0.872489	−	0.488634i	$-0.837495\pi$
$181$	− 176.885i	− 0.977268i	0.872489	−	0.488634i	$-0.162505\pi$
$182$	0	0
$183$	1.47524i	0.00806143i
$184$	0	0
$185$	43.1672	0.233336
$186$	0	0
$187$	79.2786	0.423950
$188$	0	0
$189$	25.1672i	0.133160i
$190$	0	0
$191$	231.967i	1.21449i	0.794515	+	0.607245i	$0.207725\pi$
$191$	231.967i	1.21449i	−0.794515	+	0.607245i	$0.792275\pi$
$192$	0	0
$193$	168.387	0.872471	0.436236	−	0.899832i	$-0.356311\pi$
$193$	168.387	0.872471	0.436236	+	0.899832i	$0.356311\pi$
$194$	0	0
$195$	−14.9706	−0.0767722
$196$	0	0
$197$	− 31.0820i	− 0.157777i	−0.996883	−	0.0788884i	$-0.974863\pi$
$197$	− 31.0820i	− 0.157777i	0.996883	−	0.0788884i	$-0.0251371\pi$
$198$	0	0
$199$	− 123.777i	− 0.621995i	−0.950411	−	0.310998i	$-0.899337\pi$
$199$	− 123.777i	− 0.621995i	0.950411	−	0.310998i	$-0.100663\pi$
$200$	0	0
$201$	−6.19660	−0.0308289
$202$	0	0
$203$	26.0263	0.128208
$204$	0	0
$205$	130.000i	0.634146i
$206$	0	0
$207$	142.902i	0.690346i
$208$	0	0
$209$	−341.220	−1.63263
$210$	0	0
$211$	−34.2492	−0.162319	−0.0811593	−	0.996701i	$-0.525862\pi$
$211$	−34.2492	−0.162319	−0.0811593	+	0.996701i	$0.525862\pi$
$212$	0	0
$213$	104.276i	0.489557i
$214$	0	0
$215$	− 140.344i	− 0.652765i
$216$	0	0
$217$	−42.6099	−0.196359
$218$	0	0
$219$	−87.4164	−0.399162
$220$	0	0
$221$	− 37.6130i	− 0.170195i
$222$	0	0
$223$	− 202.987i	− 0.910255i	−0.890426	−	0.455127i	$-0.849594\pi$
$223$	− 202.987i	− 0.910255i	0.890426	−	0.455127i	$-0.150406\pi$
$224$	0	0
$225$	37.3607	0.166047
$226$	0	0
$227$	−232.207	−1.02294	−0.511468	−	0.859302i	$-0.670898\pi$
$227$	−232.207	−1.02294	−0.511468	+	0.859302i	$0.670898\pi$
$228$	0	0
$229$	− 163.390i	− 0.713494i	−0.934201	−	0.356747i	$-0.883886\pi$
$229$	− 163.390i	− 0.713494i	0.934201	−	0.356747i	$-0.116114\pi$
$230$	0	0
$231$	− 17.4427i	− 0.0755096i
$232$	0	0
$233$	11.2198	0.0481537	0.0240768	−	0.999710i	$-0.492335\pi$
$233$	11.2198	0.0481537	0.0240768	+	0.999710i	$0.492335\pi$
$234$	0	0
$235$	−141.957	−0.604074
$236$	0	0
$237$	153.889i	0.649319i
$238$	0	0
$239$	216.721i	0.906784i	0.891311	+	0.453392i	$0.149786\pi$
$239$	216.721i	0.906784i	−0.891311	+	0.453392i	$0.850214\pi$
$240$	0	0
$241$	−49.1409	−0.203904	−0.101952	−	0.994789i	$-0.532509\pi$
$241$	−49.1409	−0.203904	−0.101952	+	0.994789i	$0.532509\pi$
$242$	0	0
$243$	−235.262	−0.968158
$244$	0	0
$245$	106.151i	0.433269i
$246$	0	0
$247$	161.889i	0.655419i
$248$	0	0
$249$	−198.577	−0.797500
$250$	0	0
$251$	282.525	1.12560	0.562798	−	0.826594i	$-0.309725\pi$
$251$	282.525	1.12560	0.562798	+	0.826594i	$0.309725\pi$
$252$	0	0
$253$	− 218.334i	− 0.862982i
$254$	0	0
$255$	− 19.1935i	− 0.0752686i
$256$	0	0
$257$	199.941	0.777981	0.388991	−	0.921242i	$-0.372824\pi$
$257$	199.941	0.777981	0.388991	+	0.921242i	$0.372824\pi$
$258$	0	0
$259$	23.8622	0.0921322
$260$	0	0
$261$	157.331i	0.602802i
$262$	0	0
$263$	247.761i	0.942056i	0.882118	+	0.471028i	$0.156117\pi$
$263$	247.761i	0.942056i	−0.882118	+	0.471028i	$0.843883\pi$
$264$	0	0
$265$	219.443	0.828086
$266$	0	0
$267$	57.1347	0.213987
$268$	0	0
$269$	− 13.4164i	− 0.0498751i	−0.999689	−	0.0249376i	$-0.992061\pi$
$269$	− 13.4164i	− 0.0498751i	0.999689	−	0.0249376i	$-0.00793870\pi$
$270$	0	0
$271$	− 59.4690i	− 0.219443i	−0.993962	−	0.109721i	$-0.965004\pi$
$271$	− 59.4690i	− 0.219443i	0.993962	−	0.109721i	$-0.0349959\pi$
$272$	0	0
$273$	−8.27553	−0.0303133
$274$	0	0
$275$	−57.0820	−0.207571
$276$	0	0
$277$	510.859i	1.84426i	0.386884	+	0.922128i	$0.373551\pi$
$277$	510.859i	1.84426i	−0.386884	+	0.922128i	$0.626449\pi$
$278$	0	0
$279$	− 257.580i	− 0.923228i
$280$	0	0
$281$	−489.299	−1.74128	−0.870638	−	0.491924i	$-0.836294\pi$
$281$	−489.299	−1.74128	−0.870638	+	0.491924i	$0.836294\pi$
$282$	0	0
$283$	210.233	0.742873	0.371436	−	0.928458i	$-0.378865\pi$
$283$	210.233	0.742873	0.371436	+	0.928458i	$0.378865\pi$
$284$	0	0
$285$	82.6099i	0.289859i
$286$	0	0
$287$	71.8622i	0.250391i
$288$	0	0
$289$	−240.777	−0.833139
$290$	0	0
$291$	−164.807	−0.566345
$292$	0	0
$293$	208.472i	0.711509i	0.934579	+	0.355754i	$0.115776\pi$
$293$	208.472i	0.711509i	−0.934579	+	0.355754i	$0.884224\pi$
$294$	0	0
$295$	43.1084i	0.146130i
$296$	0	0
$297$	232.446	0.782646
$298$	0	0
$299$	−103.587	−0.346444
$300$	0	0
$301$	− 77.5805i	− 0.257742i
$302$	0	0
$303$	61.5279i	0.203062i
$304$	0	0
$305$	2.66874	0.00874996
$306$	0	0
$307$	229.564	0.747766	0.373883	−	0.927476i	$-0.378026\pi$
$307$	229.564	0.747766	0.373883	+	0.927476i	$0.378026\pi$
$308$	0	0
$309$	− 243.141i	− 0.786864i
$310$	0	0
$311$	351.692i	1.13084i	0.824802	+	0.565421i	$0.191286\pi$
$311$	351.692i	1.13084i	−0.824802	+	0.565421i	$0.808714\pi$
$312$	0	0
$313$	−418.551	−1.33722	−0.668612	−	0.743612i	$-0.733111\pi$
$313$	−418.551	−1.33722	−0.668612	+	0.743612i	$0.733111\pi$
$314$	0	0
$315$	20.6525	0.0655634
$316$	0	0
$317$	351.082i	1.10751i	0.832678	+	0.553757i	$0.186806\pi$
$317$	351.082i	1.10751i	−0.832678	+	0.553757i	$0.813194\pi$
$318$	0	0
$319$	− 240.381i	− 0.753545i
$320$	0	0
$321$	18.4195	0.0573817
$322$	0	0
$323$	−207.554	−0.642583
$324$	0	0
$325$	27.0820i	0.0833294i
$326$	0	0
$327$	216.138i	0.660972i
$328$	0	0
$329$	−78.4721	−0.238517
$330$	0	0
$331$	−630.853	−1.90590	−0.952950	−	0.303127i	$-0.901969\pi$
$331$	−630.853	−1.90590	−0.952950	+	0.303127i	$0.901969\pi$
$332$	0	0
$333$	144.249i	0.433181i
$334$	0	0
$335$	11.2098i	0.0334620i
$336$	0	0
$337$	284.780	0.845045	0.422523	−	0.906352i	$-0.361145\pi$
$337$	284.780	0.845045	0.422523	+	0.906352i	$0.361145\pi$
$338$	0	0
$339$	40.8591	0.120528
$340$	0	0
$341$	393.548i	1.15410i
$342$	0	0
$343$	119.246i	0.347656i
$344$	0	0
$345$	−52.8591	−0.153215
$346$	0	0
$347$	−118.902	−0.342656	−0.171328	−	0.985214i	$-0.554806\pi$
$347$	−118.902	−0.342656	−0.171328	+	0.985214i	$0.554806\pi$
$348$	0	0
$349$	19.0495i	0.0545831i	0.999628	+	0.0272916i	$0.00868826\pi$
$349$	19.0495i	0.0545831i	−0.999628	+	0.0272916i	$0.991312\pi$
$350$	0	0
$351$	− 110.282i	− 0.314193i
$352$	0	0
$353$	−96.3344	−0.272902	−0.136451	−	0.990647i	$-0.543570\pi$
$353$	−96.3344	−0.272902	−0.136451	+	0.990647i	$0.543570\pi$
$354$	0	0
$355$	188.636	0.531370
$356$	0	0
$357$	− 10.6099i	− 0.0297196i
$358$	0	0
$359$	15.0031i	0.0417914i	0.999782	+	0.0208957i	$0.00665179\pi$
$359$	15.0031i	0.0417914i	−0.999782	+	0.0208957i	$0.993348\pi$
$360$	0	0
$361$	532.325	1.47458
$362$	0	0
$363$	−11.5379	−0.0317849
$364$	0	0
$365$	158.138i	0.433254i
$366$	0	0
$367$	− 439.348i	− 1.19713i	−0.801073	−	0.598566i	$-0.795737\pi$
$367$	− 439.348i	− 1.19713i	0.801073	−	0.598566i	$-0.204263\pi$
$368$	0	0
$369$	−434.413	−1.17727
$370$	0	0
$371$	121.305	0.326968
$372$	0	0
$373$	− 346.689i	− 0.929461i	−0.885452	−	0.464730i	$-0.846151\pi$
$373$	− 346.689i	− 0.929461i	0.885452	−	0.464730i	$-0.153849\pi$
$374$	0	0
$375$	13.8197i	0.0368524i
$376$	0	0
$377$	−114.046	−0.302510
$378$	0	0
$379$	−338.768	−0.893846	−0.446923	−	0.894572i	$-0.647480\pi$
$379$	−338.768	−0.893846	−0.446923	+	0.894572i	$0.647480\pi$
$380$	0	0
$381$	115.862i	0.304100i
$382$	0	0
$383$	142.351i	0.371673i	0.982581	+	0.185836i	$0.0594994\pi$
$383$	142.351i	0.371673i	−0.982581	+	0.185836i	$0.940501\pi$
$384$	0	0
$385$	−31.5542	−0.0819589
$386$	0	0
$387$	468.981	1.21184
$388$	0	0
$389$	91.6331i	0.235561i	0.993040	+	0.117780i	$0.0375779\pi$
$389$	91.6331i	0.235561i	−0.993040	+	0.117780i	$0.962422\pi$
$390$	0	0
$391$	− 132.807i	− 0.339659i
$392$	0	0
$393$	99.8823	0.254154
$394$	0	0
$395$	278.387	0.704777
$396$	0	0
$397$	384.354i	0.968147i	0.875027	+	0.484074i	$0.160843\pi$
$397$	384.354i	0.968147i	−0.875027	+	0.484074i	$0.839157\pi$
$398$	0	0
$399$	45.6656i	0.114450i
$400$	0	0
$401$	−416.545	−1.03877	−0.519383	−	0.854542i	$-0.673838\pi$
$401$	−416.545	−1.03877	−0.519383	+	0.854542i	$0.673838\pi$
$402$	0	0
$403$	186.715	0.463313
$404$	0	0
$405$	94.0983i	0.232341i
$406$	0	0
$407$	− 220.393i	− 0.541507i
$408$	0	0
$409$	−764.354	−1.86884	−0.934419	−	0.356177i	$-0.884080\pi$
$409$	−764.354	−1.86884	−0.934419	+	0.356177i	$0.884080\pi$
$410$	0	0
$411$	−237.528	−0.577927
$412$	0	0
$413$	23.8297i	0.0576991i
$414$	0	0
$415$	359.230i	0.865614i
$416$	0	0
$417$	8.17029	0.0195930
$418$	0	0
$419$	117.613	0.280699	0.140350	−	0.990102i	$-0.455177\pi$
$419$	117.613	0.280699	0.140350	+	0.990102i	$0.455177\pi$
$420$	0	0
$421$	− 429.915i	− 1.02118i	−0.859826	−	0.510588i	$-0.829428\pi$
$421$	− 429.915i	− 1.02118i	0.859826	−	0.510588i	$-0.170572\pi$
$422$	0	0
$423$	− 474.371i	− 1.12144i
$424$	0	0
$425$	−34.7214	−0.0816973
$426$	0	0
$427$	1.47524	0.00345490
$428$	0	0
$429$	76.4334i	0.178166i
$430$	0	0
$431$	− 596.466i	− 1.38391i	−0.721940	−	0.691956i	$-0.756749\pi$
$431$	− 596.466i	− 1.38391i	0.721940	−	0.691956i	$-0.243251\pi$
$432$	0	0
$433$	−678.551	−1.56709	−0.783546	−	0.621333i	$-0.786591\pi$
$433$	−678.551	−1.56709	−0.783546	+	0.621333i	$0.786591\pi$
$434$	0	0
$435$	−58.1966	−0.133785
$436$	0	0
$437$	571.607i	1.30802i
$438$	0	0
$439$	324.774i	0.739804i	0.929071	+	0.369902i	$0.120609\pi$
$439$	324.774i	0.739804i	−0.929071	+	0.369902i	$0.879391\pi$
$440$	0	0
$441$	−354.718	−0.804350
$442$	0	0
$443$	49.4064	0.111527	0.0557634	−	0.998444i	$-0.482241\pi$
$443$	49.4064	0.111527	0.0557634	+	0.998444i	$0.482241\pi$
$444$	0	0
$445$	− 103.358i	− 0.232264i
$446$	0	0
$447$	246.971i	0.552507i
$448$	0	0
$449$	−186.859	−0.416167	−0.208084	−	0.978111i	$-0.566723\pi$
$449$	−186.859	−0.416167	−0.208084	+	0.978111i	$0.566723\pi$
$450$	0	0
$451$	663.724	1.47167
$452$	0	0
$453$	65.8235i	0.145306i
$454$	0	0
$455$	14.9706i	0.0329024i
$456$	0	0
$457$	645.214	1.41185	0.705923	−	0.708289i	$-0.250532\pi$
$457$	645.214	1.41185	0.705923	+	0.708289i	$0.250532\pi$
$458$	0	0
$459$	141.390	0.308039
$460$	0	0
$461$	− 10.7864i	− 0.0233978i	−0.999932	−	0.0116989i	$-0.996276\pi$
$461$	− 10.7864i	− 0.0233978i	0.999932	−	0.0116989i	$-0.00372397\pi$
$462$	0	0
$463$	594.233i	1.28344i	0.766939	+	0.641720i	$0.221779\pi$
$463$	594.233i	1.28344i	−0.766939	+	0.641720i	$0.778221\pi$
$464$	0	0
$465$	95.2786	0.204900
$466$	0	0
$467$	−246.148	−0.527083	−0.263542	−	0.964648i	$-0.584891\pi$
$467$	−246.148	−0.527083	−0.263542	+	0.964648i	$0.584891\pi$
$468$	0	0
$469$	6.19660i	0.0132124i
$470$	0	0
$471$	54.1440i	0.114955i
$472$	0	0
$473$	−716.539	−1.51488
$474$	0	0
$475$	149.443	0.314616
$476$	0	0
$477$	733.299i	1.53731i
$478$	0	0
$479$	− 722.610i	− 1.50858i	−0.656541	−	0.754290i	$-0.727981\pi$
$479$	− 722.610i	− 1.50858i	0.656541	−	0.754290i	$-0.272019\pi$
$480$	0	0
$481$	−104.563	−0.217388
$482$	0	0
$483$	−29.2198	−0.0604965
$484$	0	0
$485$	298.138i	0.614717i
$486$	0	0
$487$	482.403i	0.990561i	0.868733	+	0.495281i	$0.164935\pi$
$487$	482.403i	0.990561i	−0.868733	+	0.495281i	$0.835065\pi$
$488$	0	0
$489$	−233.252	−0.476999
$490$	0	0
$491$	235.967	0.480585	0.240293	−	0.970700i	$-0.422757\pi$
$491$	235.967	0.480585	0.240293	+	0.970700i	$0.422757\pi$
$492$	0	0
$493$	− 146.217i	− 0.296586i
$494$	0	0
$495$	− 190.748i	− 0.385349i
$496$	0	0
$497$	104.276	0.209810
$498$	0	0
$499$	−591.331	−1.18503	−0.592516	−	0.805558i	$-0.701865\pi$
$499$	−591.331	−1.18503	−0.592516	+	0.805558i	$0.701865\pi$
$500$	0	0
$501$	322.249i	0.643212i
$502$	0	0
$503$	− 708.843i	− 1.40923i	−0.709590	−	0.704615i	$-0.751120\pi$
$503$	− 708.843i	− 1.40923i	0.709590	−	0.704615i	$-0.248880\pi$
$504$	0	0
$505$	111.305	0.220406
$506$	0	0
$507$	−172.632	−0.340498
$508$	0	0
$509$	− 801.489i	− 1.57463i	−0.616548	−	0.787317i	$-0.711469\pi$
$509$	− 801.489i	− 1.57463i	0.616548	−	0.787317i	$-0.288531\pi$
$510$	0	0
$511$	87.4164i	0.171069i
$512$	0	0
$513$	−608.551	−1.18626
$514$	0	0
$515$	−439.846	−0.854070
$516$	0	0
$517$	724.774i	1.40188i
$518$	0	0
$519$	35.4690i	0.0683411i
$520$	0	0
$521$	−285.882	−0.548718	−0.274359	−	0.961627i	$-0.588466\pi$
$521$	−285.882	−0.548718	−0.274359	+	0.961627i	$0.588466\pi$
$522$	0	0
$523$	493.013	0.942664	0.471332	−	0.881956i	$-0.343773\pi$
$523$	493.013	0.942664	0.471332	+	0.881956i	$0.343773\pi$
$524$	0	0
$525$	7.63932i	0.0145511i
$526$	0	0
$527$	239.384i	0.454239i
$528$	0	0
$529$	163.249	0.308600
$530$	0	0
$531$	−144.053	−0.271286
$532$	0	0
$533$	− 314.898i	− 0.590803i
$534$	0	0
$535$	− 33.3212i	− 0.0622826i
$536$	0	0
$537$	−118.046	−0.219826
$538$	0	0
$539$	541.961	1.00549
$540$	0	0
$541$	906.774i	1.67611i	0.545588	+	0.838054i	$0.316306\pi$
$541$	906.774i	1.67611i	−0.545588	+	0.838054i	$0.683694\pi$
$542$	0	0
$543$	218.642i	0.402656i
$544$	0	0
$545$	390.997	0.717425
$546$	0	0
$547$	−111.420	−0.203693	−0.101847	−	0.994800i	$-0.532475\pi$
$547$	−111.420	−0.203693	−0.101847	+	0.994800i	$0.532475\pi$
$548$	0	0
$549$	8.91796i	0.0162440i
$550$	0	0
$551$	629.325i	1.14215i
$552$	0	0
$553$	153.889	0.278279
$554$	0	0
$555$	−53.3576	−0.0961398
$556$	0	0
$557$	462.255i	0.829902i	0.909844	+	0.414951i	$0.136201\pi$
$557$	462.255i	0.829902i	−0.909844	+	0.414951i	$0.863799\pi$
$558$	0	0
$559$	339.955i	0.608149i
$560$	0	0
$561$	−97.9938	−0.174677
$562$	0	0
$563$	−640.555	−1.13775	−0.568876	−	0.822423i	$-0.692622\pi$
$563$	−640.555	−1.13775	−0.568876	+	0.822423i	$0.692622\pi$
$564$	0	0
$565$	− 73.9149i	− 0.130823i
$566$	0	0
$567$	52.0163i	0.0917394i
$568$	0	0
$569$	−817.915	−1.43746	−0.718730	−	0.695289i	$-0.755276\pi$
$569$	−817.915	−1.43746	−0.718730	+	0.695289i	$0.755276\pi$
$570$	0	0
$571$	−470.642	−0.824242	−0.412121	−	0.911129i	$-0.635212\pi$
$571$	−470.642	−0.824242	−0.412121	+	0.911129i	$0.635212\pi$
$572$	0	0
$573$	− 286.728i	− 0.500397i
$574$	0	0
$575$	95.6231i	0.166301i
$576$	0	0
$577$	1051.17	1.82178	0.910890	−	0.412649i	$-0.135396\pi$
$577$	1051.17	1.82178	0.910890	+	0.412649i	$0.135396\pi$
$578$	0	0
$579$	−208.138	−0.359478
$580$	0	0
$581$	198.577i	0.341786i
$582$	0	0
$583$	− 1120.38i	− 1.92175i
$584$	0	0
$585$	−90.4984	−0.154698
$586$	0	0
$587$	267.708	0.456062	0.228031	−	0.973654i	$-0.426771\pi$
$587$	267.708	0.456062	0.228031	+	0.973654i	$0.426771\pi$
$588$	0	0
$589$	− 1030.32i	− 1.74927i
$590$	0	0
$591$	38.4195i	0.0650076i
$592$	0	0
$593$	−168.440	−0.284047	−0.142023	−	0.989863i	$-0.545361\pi$
$593$	−168.440	−0.284047	−0.142023	+	0.989863i	$0.545361\pi$
$594$	0	0
$595$	−19.1935	−0.0322580
$596$	0	0
$597$	152.997i	0.256276i
$598$	0	0
$599$	626.715i	1.04627i	0.852250	+	0.523135i	$0.175237\pi$
$599$	626.715i	1.04627i	−0.852250	+	0.523135i	$0.824763\pi$
$600$	0	0
$601$	359.252	0.597758	0.298879	−	0.954291i	$-0.403387\pi$
$601$	359.252	0.597758	0.298879	+	0.954291i	$0.403387\pi$
$602$	0	0
$603$	−37.4590	−0.0621210
$604$	0	0
$605$	20.8723i	0.0344996i
$606$	0	0
$607$	− 679.728i	− 1.11982i	−0.828555	−	0.559908i	$-0.810836\pi$
$607$	− 679.728i	− 1.11982i	0.828555	−	0.559908i	$-0.189164\pi$
$608$	0	0
$609$	−32.1703	−0.0528248
$610$	0	0
$611$	343.862	0.562786
$612$	0	0
$613$	− 748.958i	− 1.22179i	−0.791711	−	0.610896i	$-0.790809\pi$
$613$	− 748.958i	− 1.22179i	0.791711	−	0.610896i	$-0.209191\pi$
$614$	0	0
$615$	− 160.689i	− 0.261283i
$616$	0	0
$617$	947.876	1.53627	0.768133	−	0.640290i	$-0.221186\pi$
$617$	947.876	1.53627	0.768133	+	0.640290i	$0.221186\pi$
$618$	0	0
$619$	1074.32	1.73558	0.867788	−	0.496934i	$-0.165541\pi$
$619$	1074.32	1.73558	0.867788	+	0.496934i	$0.165541\pi$
$620$	0	0
$621$	− 389.390i	− 0.627037i
$622$	0	0
$623$	− 57.1347i	− 0.0917089i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	421.771	0.672681
$628$	0	0
$629$	− 134.059i	− 0.213130i
$630$	0	0
$631$	− 1061.67i	− 1.68253i	−0.540627	−	0.841263i	$-0.681813\pi$
$631$	− 1061.67i	− 1.68253i	0.540627	−	0.841263i	$-0.318187\pi$
$632$	0	0
$633$	42.3344	0.0668789
$634$	0	0
$635$	209.597	0.330074
$636$	0	0
$637$	− 257.128i	− 0.403655i
$638$	0	0
$639$	630.354i	0.986470i
$640$	0	0
$641$	497.404	0.775981	0.387991	−	0.921663i	$-0.373169\pi$
$641$	497.404	0.775981	0.387991	+	0.921663i	$0.373169\pi$
$642$	0	0
$643$	−604.397	−0.939964	−0.469982	−	0.882676i	$-0.655740\pi$
$643$	−604.397	−0.939964	−0.469982	+	0.882676i	$0.655740\pi$
$644$	0	0
$645$	173.475i	0.268954i
$646$	0	0
$647$	− 693.151i	− 1.07133i	−0.844430	−	0.535665i	$-0.820061\pi$
$647$	− 693.151i	− 1.07133i	0.844430	−	0.535665i	$-0.179939\pi$
$648$	0	0
$649$	220.093	0.339126
$650$	0	0
$651$	52.6687	0.0809044
$652$	0	0
$653$	948.853i	1.45307i	0.687131	+	0.726534i	$0.258870\pi$
$653$	948.853i	1.45307i	−0.687131	+	0.726534i	$0.741130\pi$
$654$	0	0
$655$	− 180.689i	− 0.275861i
$656$	0	0
$657$	−528.440	−0.804322
$658$	0	0
$659$	889.260	1.34941	0.674704	−	0.738088i	$-0.264271\pi$
$659$	889.260	1.34941	0.674704	+	0.738088i	$0.264271\pi$
$660$	0	0
$661$	− 664.354i	− 1.00507i	−0.864555	−	0.502537i	$-0.832400\pi$
$661$	− 664.354i	− 1.00507i	0.864555	−	0.502537i	$-0.167600\pi$
$662$	0	0
$663$	46.4922i	0.0701240i
$664$	0	0
$665$	82.6099	0.124225
$666$	0	0
$667$	−402.683	−0.603722
$668$	0	0
$669$	250.906i	0.375046i
$670$	0	0
$671$	− 13.6254i	− 0.0203062i
$672$	0	0
$673$	429.437	0.638093	0.319046	−	0.947739i	$-0.396637\pi$
$673$	429.437	0.638093	0.319046	+	0.947739i	$0.396637\pi$
$674$	0	0
$675$	−101.803	−0.150820
$676$	0	0
$677$	336.918i	0.497663i	0.968547	+	0.248832i	$0.0800466\pi$
$677$	336.918i	0.497663i	−0.968547	+	0.248832i	$0.919953\pi$
$678$	0	0
$679$	164.807i	0.242719i
$680$	0	0
$681$	287.023	0.421473
$682$	0	0
$683$	−818.200	−1.19795	−0.598975	−	0.800767i	$-0.704425\pi$
$683$	−818.200	−1.19795	−0.598975	+	0.800767i	$0.704425\pi$
$684$	0	0
$685$	429.692i	0.627288i
$686$	0	0
$687$	201.961i	0.293976i
$688$	0	0
$689$	−531.554	−0.771486
$690$	0	0
$691$	356.689	0.516192	0.258096	−	0.966119i	$-0.416905\pi$
$691$	356.689	0.516192	0.258096	+	0.966119i	$0.416905\pi$
$692$	0	0
$693$	− 105.443i	− 0.152154i
$694$	0	0
$695$	− 14.7802i	− 0.0212665i
$696$	0	0
$697$	403.724	0.579232
$698$	0	0
$699$	−13.8684	−0.0198404
$700$	0	0
$701$	− 50.0851i	− 0.0714481i	−0.999362	−	0.0357241i	$-0.988626\pi$
$701$	− 50.0851i	− 0.0714481i	0.999362	−	0.0357241i	$-0.0113737\pi$
$702$	0	0
$703$	576.997i	0.820764i
$704$	0	0
$705$	175.469	0.248892
$706$	0	0
$707$	61.5279	0.0870267
$708$	0	0
$709$	− 698.604i	− 0.985337i	−0.870217	−	0.492668i	$-0.836022\pi$
$709$	− 698.604i	− 0.985337i	0.870217	−	0.492668i	$-0.163978\pi$
$710$	0	0
$711$	930.269i	1.30840i
$712$	0	0
$713$	659.266	0.924637
$714$	0	0
$715$	138.269	0.193384
$716$	0	0
$717$	− 267.882i	− 0.373616i
$718$	0	0
$719$	− 246.387i	− 0.342680i	−0.985212	−	0.171340i	$-0.945190\pi$
$719$	− 246.387i	− 0.342680i	0.985212	−	0.171340i	$-0.0548097\pi$
$720$	0	0
$721$	−243.141	−0.337227
$722$	0	0
$723$	60.7415	0.0840131
$724$	0	0
$725$	105.279i	0.145212i
$726$	0	0
$727$	256.134i	0.352316i	0.984362	+	0.176158i	$0.0563670\pi$
$727$	256.134i	0.352316i	−0.984362	+	0.176158i	$0.943633\pi$
$728$	0	0
$729$	−87.9381	−0.120628
$730$	0	0
$731$	−435.850	−0.596238
$732$	0	0
$733$	− 897.574i	− 1.22452i	−0.790656	−	0.612261i	$-0.790260\pi$
$733$	− 897.574i	− 1.22452i	0.790656	−	0.612261i	$-0.209740\pi$
$734$	0	0
$735$	− 131.210i	− 0.178517i
$736$	0	0
$737$	57.2322	0.0776557
$738$	0	0
$739$	268.486	0.363310	0.181655	−	0.983362i	$-0.441855\pi$
$739$	268.486	0.363310	0.181655	+	0.983362i	$0.441855\pi$
$740$	0	0
$741$	− 200.105i	− 0.270048i
$742$	0	0
$743$	612.049i	0.823753i	0.911240	+	0.411877i	$0.135127\pi$
$743$	612.049i	0.823753i	−0.911240	+	0.411877i	$0.864873\pi$
$744$	0	0
$745$	446.774	0.599697
$746$	0	0
$747$	−1200.42	−1.60698
$748$	0	0
$749$	− 18.4195i	− 0.0245921i
$750$	0	0
$751$	142.485i	0.189726i	0.995490	+	0.0948632i	$0.0302414\pi$
$751$	142.485i	0.189726i	−0.995490	+	0.0948632i	$0.969759\pi$
$752$	0	0
$753$	−349.220	−0.463771
$754$	0	0
$755$	119.076	0.157716
$756$	0	0
$757$	− 636.342i	− 0.840610i	−0.907383	−	0.420305i	$-0.861923\pi$
$757$	− 636.342i	− 0.840610i	0.907383	−	0.420305i	$-0.138077\pi$
$758$	0	0
$759$	269.876i	0.355568i
$760$	0	0
$761$	−591.666	−0.777484	−0.388742	−	0.921347i	$-0.627090\pi$
$761$	−591.666	−0.777484	−0.388742	+	0.921347i	$0.627090\pi$
$762$	0	0
$763$	216.138	0.283274
$764$	0	0
$765$	− 116.026i	− 0.151668i
$766$	0	0
$767$	− 104.421i	− 0.136142i
$768$	0	0
$769$	0.216701	0.000281796 0	0.000140898	−	1.00000i	$-0.499955\pi$
$769$	0.216701	0.000281796 0	0.000140898		1.00000i	$0.499955\pi$
$770$	0	0
$771$	−247.141	−0.320546
$772$	0	0
$773$	762.807i	0.986813i	0.869799	+	0.493407i	$0.164249\pi$
$773$	762.807i	0.986813i	−0.869799	+	0.493407i	$0.835751\pi$
$774$	0	0
$775$	− 172.361i	− 0.222401i
$776$	0	0
$777$	−29.4953	−0.0379605
$778$	0	0
$779$	−1737.65	−2.23062
$780$	0	0
$781$	− 963.096i	− 1.23316i
$782$	0	0
$783$	− 428.709i	− 0.547521i
$784$	0	0
$785$	97.9474	0.124774
$786$	0	0
$787$	−1311.80	−1.66684	−0.833419	−	0.552642i	$-0.813620\pi$
$787$	−1311.80	−1.66684	−0.833419	+	0.552642i	$0.813620\pi$
$788$	0	0
$789$	− 306.249i	− 0.388149i
$790$	0	0
$791$	− 40.8591i	− 0.0516550i
$792$	0	0
$793$	−6.46446	−0.00815190
$794$	0	0
$795$	−271.246	−0.341190
$796$	0	0
$797$	1341.85i	1.68363i	0.539769	+	0.841813i	$0.318511\pi$
$797$	1341.85i	1.68363i	−0.539769	+	0.841813i	$0.681489\pi$
$798$	0	0
$799$	440.859i	0.551764i
$800$	0	0
$801$	345.384	0.431191
$802$	0	0
$803$	807.384	1.00546
$804$	0	0
$805$	52.8591i	0.0656635i
$806$	0	0
$807$	16.5836i	0.0205497i
$808$	0	0
$809$	53.8948	0.0666190	0.0333095	−	0.999445i	$-0.489395\pi$
$809$	53.8948	0.0666190	0.0333095	+	0.999445i	$0.489395\pi$
$810$	0	0
$811$	20.9644	0.0258500	0.0129250	−	0.999916i	$-0.495886\pi$
$811$	20.9644	0.0258500	0.0129250	+	0.999916i	$0.495886\pi$
$812$	0	0
$813$	73.5078i	0.0904155i
$814$	0	0
$815$	421.957i	0.517739i
$816$	0	0
$817$	1875.92	2.29611
$818$	0	0
$819$	−50.0263	−0.0610822
$820$	0	0
$821$	− 363.240i	− 0.442436i	−0.975224	−	0.221218i	$-0.928997\pi$
$821$	− 363.240i	− 0.442436i	0.975224	−	0.221218i	$-0.0710032\pi$
$822$	0	0
$823$	1487.42i	1.80732i	0.428256	+	0.903658i	$0.359128\pi$
$823$	1487.42i	1.80732i	−0.428256	+	0.903658i	$0.640872\pi$
$824$	0	0
$825$	70.5573	0.0855240
$826$	0	0
$827$	948.981	1.14750	0.573749	−	0.819031i	$-0.305489\pi$
$827$	948.981	1.14750	0.573749	+	0.819031i	$0.305489\pi$
$828$	0	0
$829$	− 1114.06i	− 1.34386i	−0.740614	−	0.671930i	$-0.765465\pi$
$829$	− 1114.06i	− 1.34386i	0.740614	−	0.671930i	$-0.234535\pi$
$830$	0	0
$831$	− 631.457i	− 0.759876i
$832$	0	0
$833$	329.659	0.395750
$834$	0	0
$835$	582.954	0.698149
$836$	0	0
$837$	701.876i	0.838562i
$838$	0	0
$839$	145.272i	0.173149i	0.996245	+	0.0865747i	$0.0275921\pi$
$839$	145.272i	0.173149i	−0.996245	+	0.0865747i	$0.972408\pi$
$840$	0	0
$841$	397.656	0.472837
$842$	0	0
$843$	604.807	0.717445
$844$	0	0
$845$	312.295i	0.369580i
$846$	0	0
$847$	11.5379i	0.0136221i
$848$	0	0
$849$	−259.862	−0.306080
$850$	0	0
$851$	−369.200	−0.433842
$852$	0	0
$853$	− 127.423i	− 0.149382i	−0.997207	−	0.0746909i	$-0.976203\pi$
$853$	− 127.423i	− 0.149382i	0.997207	−	0.0746909i	$-0.0237970\pi$
$854$	0	0
$855$	499.384i	0.584075i
$856$	0	0
$857$	576.715	0.672946	0.336473	−	0.941693i	$-0.390766\pi$
$857$	576.715	0.672946	0.336473	+	0.941693i	$0.390766\pi$
$858$	0	0
$859$	513.155	0.597386	0.298693	−	0.954349i	$-0.403449\pi$
$859$	513.155	0.597386	0.298693	+	0.954349i	$0.403449\pi$
$860$	0	0
$861$	− 88.8266i	− 0.103167i
$862$	0	0
$863$	− 774.488i	− 0.897437i	−0.893673	−	0.448719i	$-0.851881\pi$
$863$	− 774.488i	− 0.897437i	0.893673	−	0.448719i	$-0.148119\pi$
$864$	0	0
$865$	64.1641	0.0741781
$866$	0	0
$867$	297.617	0.343272
$868$	0	0
$869$	− 1421.33i	− 1.63559i
$870$	0	0
$871$	− 27.1533i	− 0.0311749i
$872$	0	0
$873$	−996.269	−1.14120
$874$	0	0
$875$	13.8197	0.0157939
$876$	0	0
$877$	− 1335.13i	− 1.52239i	−0.648524	−	0.761194i	$-0.724613\pi$
$877$	− 1335.13i	− 1.52239i	0.648524	−	0.761194i	$-0.275387\pi$
$878$	0	0
$879$	− 257.686i	− 0.293158i
$880$	0	0
$881$	29.7570	0.0337764	0.0168882	−	0.999857i	$-0.494624\pi$
$881$	29.7570	0.0337764	0.0168882	+	0.999857i	$0.494624\pi$
$882$	0	0
$883$	105.544	0.119529	0.0597645	−	0.998213i	$-0.480965\pi$
$883$	105.544	0.119529	0.0597645	+	0.998213i	$0.480965\pi$
$884$	0	0
$885$	− 53.2849i	− 0.0602089i
$886$	0	0
$887$	60.2268i	0.0678994i	0.999424	+	0.0339497i	$0.0108086\pi$
$887$	60.2268i	0.0678994i	−0.999424	+	0.0339497i	$0.989191\pi$
$888$	0	0
$889$	115.862	0.130329
$890$	0	0
$891$	480.426	0.539198
$892$	0	0
$893$	− 1897.48i	− 2.12484i
$894$	0	0
$895$	213.548i	0.238601i
$896$	0	0
$897$	128.040	0.142743
$898$	0	0
$899$	725.836	0.807381
$900$	0	0
$901$	− 681.495i	− 0.756377i
$902$	0	0
$903$	95.8948i	0.106196i
$904$	0	0
$905$	395.528	0.437047
$906$	0	0
$907$	155.473	0.171414	0.0857072	−	0.996320i	$-0.472685\pi$
$907$	155.473	0.171414	0.0857072	+	0.996320i	$0.472685\pi$
$908$	0	0
$909$	371.941i	0.409176i
$910$	0	0
$911$	− 686.630i	− 0.753710i	−0.926272	−	0.376855i	$-0.877005\pi$
$911$	− 686.630i	− 0.753710i	0.926272	−	0.376855i	$-0.122995\pi$
$912$	0	0
$913$	1834.07	2.00884
$914$	0	0
$915$	−3.29874	−0.00360518
$916$	0	0
$917$	− 99.8823i	− 0.108923i
$918$	0	0
$919$	− 1339.16i	− 1.45719i	−0.684943	−	0.728597i	$-0.740173\pi$
$919$	− 1339.16i	− 1.45719i	0.684943	−	0.728597i	$-0.259827\pi$
$920$	0	0
$921$	−283.757	−0.308097
$922$	0	0
$923$	−456.932	−0.495051
$924$	0	0
$925$	96.5248i	0.104351i
$926$	0	0
$927$	− 1469.81i	− 1.58555i
$928$	0	0
$929$	−636.302	−0.684932	−0.342466	−	0.939530i	$-0.611262\pi$
$929$	−636.302	−0.684932	−0.342466	+	0.939530i	$0.611262\pi$
$930$	0	0
$931$	−1418.87	−1.52403
$932$	0	0
$933$	− 434.715i	− 0.465933i
$934$	0	0
$935$	177.272i	0.189596i
$936$	0	0
$937$	310.210	0.331068	0.165534	−	0.986204i	$-0.447065\pi$
$937$	310.210	0.331068	0.165534	+	0.986204i	$0.447065\pi$
$938$	0	0
$939$	517.358	0.550967
$940$	0	0
$941$	− 1322.66i	− 1.40559i	−0.711394	−	0.702793i	$-0.751936\pi$
$941$	− 1322.66i	− 1.40559i	0.711394	−	0.702793i	$-0.248064\pi$
$942$	0	0
$943$	− 1111.86i	− 1.17907i
$944$	0	0
$945$	−56.2755	−0.0595508
$946$	0	0
$947$	724.883	0.765452	0.382726	−	0.923862i	$-0.374985\pi$
$947$	724.883	0.765452	0.382726	+	0.923862i	$0.374985\pi$
$948$	0	0
$949$	− 383.056i	− 0.403641i
$950$	0	0
$951$	− 433.961i	− 0.456321i
$952$	0	0
$953$	298.223	0.312931	0.156465	−	0.987683i	$-0.449990\pi$
$953$	298.223	0.312931	0.156465	+	0.987683i	$0.449990\pi$
$954$	0	0
$955$	−518.695	−0.543136
$956$	0	0
$957$	297.127i	0.310478i
$958$	0	0
$959$	237.528i	0.247683i
$960$	0	0
$961$	−227.328	−0.236554
$962$	0	0
$963$	111.348	0.115626
$964$	0	0
$965$	376.525i	0.390181i
$966$	0	0
$967$	249.236i	0.257742i	0.991661	+	0.128871i	$0.0411352\pi$
$967$	249.236i	0.257742i	−0.991661	+	0.128871i	$0.958865\pi$
$968$	0	0
$969$	256.551	0.264759
$970$	0	0
$971$	339.246	0.349378	0.174689	−	0.984624i	$-0.444108\pi$
$971$	339.246	0.349378	0.174689	+	0.984624i	$0.444108\pi$
$972$	0	0
$973$	− 8.17029i	− 0.00839701i
$974$	0	0
$975$	− 33.4752i	− 0.0343336i
$976$	0	0
$977$	−40.8854	−0.0418479	−0.0209240	−	0.999781i	$-0.506661\pi$
$977$	−40.8854	−0.0418479	−0.0209240	+	0.999781i	$0.506661\pi$
$978$	0	0
$979$	−527.700	−0.539019
$980$	0	0
$981$	1306.57i	1.33188i
$982$	0	0
$983$	− 5.56423i	− 0.00566045i	−0.999996	−	0.00283023i	$-0.999099\pi$
$983$	− 5.56423i	− 0.00566045i	0.999996	−	0.00283023i	$-0.000900890\pi$
$984$	0	0
$985$	69.5016	0.0705600
$986$	0	0
$987$	96.9969	0.0982745
$988$	0	0
$989$	1200.34i	1.21369i
$990$	0	0
$991$	1562.45i	1.57664i	0.615267	+	0.788319i	$0.289048\pi$
$991$	1562.45i	1.57664i	−0.615267	+	0.788319i	$0.710952\pi$
$992$	0	0
$993$	779.777	0.785274
$994$	0	0
$995$	276.774	0.278165
$996$	0	0
$997$	− 704.224i	− 0.706343i	−0.935559	−	0.353172i	$-0.885103\pi$
$997$	− 704.224i	− 0.706343i	0.935559	−	0.353172i	$-0.114897\pi$
$998$	0	0
$999$	− 393.062i	− 0.393455i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.g.c.1151.2		4
4.3	odd	2		1280.3.g.b.1151.4		4
8.3	odd	2	inner	1280.3.g.c.1151.1		4
8.5	even	2		1280.3.g.b.1151.3		4
16.3	odd	4		160.3.b.b.31.3	yes	4
16.5	even	4		320.3.b.d.191.3		4
16.11	odd	4		320.3.b.d.191.2		4
16.13	even	4		160.3.b.b.31.2	✓	4
48.5	odd	4		2880.3.e.h.2431.3		4
48.11	even	4		2880.3.e.h.2431.4		4
48.29	odd	4		1440.3.e.a.991.1		4
48.35	even	4		1440.3.e.a.991.2		4
80.3	even	4		800.3.h.h.799.2		4
80.13	odd	4		800.3.h.e.799.3		4
80.19	odd	4		800.3.b.g.351.2		4
80.27	even	4		1600.3.h.k.1599.1		4
80.29	even	4		800.3.b.g.351.3		4
80.37	odd	4		1600.3.h.f.1599.4		4
80.43	even	4		1600.3.h.f.1599.3		4
80.53	odd	4		1600.3.h.k.1599.2		4
80.59	odd	4		1600.3.b.u.1151.3		4
80.67	even	4		800.3.h.e.799.4		4
80.69	even	4		1600.3.b.u.1151.2		4
80.77	odd	4		800.3.h.h.799.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.b.31.2	✓	4	16.13	even	4
160.3.b.b.31.3	yes	4	16.3	odd	4
320.3.b.d.191.2		4	16.11	odd	4
320.3.b.d.191.3		4	16.5	even	4
800.3.b.g.351.2		4	80.19	odd	4
800.3.b.g.351.3		4	80.29	even	4
800.3.h.e.799.3		4	80.13	odd	4
800.3.h.e.799.4		4	80.67	even	4
800.3.h.h.799.1		4	80.77	odd	4
800.3.h.h.799.2		4	80.3	even	4
1280.3.g.b.1151.3		4	8.5	even	2
1280.3.g.b.1151.4		4	4.3	odd	2
1280.3.g.c.1151.1		4	8.3	odd	2	inner
1280.3.g.c.1151.2		4	1.1	even	1	trivial
1440.3.e.a.991.1		4	48.29	odd	4
1440.3.e.a.991.2		4	48.35	even	4
1600.3.b.u.1151.2		4	80.69	even	4
1600.3.b.u.1151.3		4	80.59	odd	4
1600.3.h.f.1599.3		4	80.43	even	4
1600.3.h.f.1599.4		4	80.37	odd	4
1600.3.h.k.1599.1		4	80.27	even	4
1600.3.h.k.1599.2		4	80.53	odd	4
2880.3.e.h.2431.3		4	48.5	odd	4
2880.3.e.h.2431.4		4	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.23607 q^{3} +2.23607i q^{5} +1.23607i q^{7} -7.47214 q^{9} +11.4164 q^{11} -5.41641i q^{13} -2.76393i q^{15} +6.94427 q^{17} -29.8885 q^{19} -1.52786i q^{21} -19.1246i q^{23} -5.00000 q^{25} +20.3607 q^{27} -21.0557i q^{29} +34.4721i q^{31} -14.1115 q^{33} -2.76393 q^{35} -19.3050i q^{37} +6.69505i q^{39} +58.1378 q^{41} -62.7639 q^{43} -16.7082i q^{45} +63.4853i q^{47} +47.4721 q^{49} -8.58359 q^{51} -98.1378i q^{53} +25.5279i q^{55} +36.9443 q^{57} +19.2786 q^{59} -1.19350i q^{61} -9.23607i q^{63} +12.1115 q^{65} +5.01316 q^{67} +23.6393i q^{69} -84.3607i q^{71} +70.7214 q^{73} +6.18034 q^{75} +14.1115i q^{77} -124.498i q^{79} +42.0820 q^{81} +160.652 q^{83} +15.5279i q^{85} +26.0263i q^{87} -46.2229 q^{89} +6.69505 q^{91} -42.6099i q^{93} -66.8328i q^{95} +133.331 q^{97} -85.3050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 12 q^{9} - 8 q^{11} - 8 q^{17} - 48 q^{19} - 20 q^{25} - 8 q^{27} - 128 q^{33} - 20 q^{35} - 260 q^{43} + 172 q^{49} - 88 q^{51} + 112 q^{57} + 256 q^{59} + 120 q^{65} - 132 q^{67} + 104 q^{73}+ \cdots - 216 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 12 * q^9 - 8 * q^11 - 8 * q^17 - 48 * q^19 - 20 * q^25 - 8 * q^27 - 128 * q^33 - 20 * q^35 - 260 * q^43 + 172 * q^49 - 88 * q^51 + 112 * q^57 + 256 * q^59 + 120 * q^65 - 132 * q^67 + 104 * q^73 - 20 * q^75 - 100 * q^81 + 580 * q^83 - 328 * q^89 + 152 * q^91 + 104 * q^97 - 216 * q^99

Introduction

L-functions

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