LMFDB - Embedded newform 1600.3.b.w.1151.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1151,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.b (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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.5
Root			$1.37720i$ of defining polynomial
Character	$\chi$	$=$	1600.1151
Dual form			1600.3.b.w.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.75441i q^{3} +3.84997i q^{7} +1.41325 q^{9} +O(q^{10})$	$q+2.75441i q^{3} +3.84997i q^{7} +1.41325 q^{9} +6.19112i q^{11} -16.1132 q^{13} -5.20875 q^{17} -36.2264i q^{19} -10.6044 q^{21} -22.0411i q^{23} +28.6823i q^{27} -20.0352 q^{29} +26.4175i q^{31} -17.0529 q^{33} -69.3219 q^{37} -44.3822i q^{39} +11.6220 q^{41} -25.8542i q^{43} -66.1853i q^{47} +34.1777 q^{49} -14.3470i q^{51} -39.5751 q^{53} +99.7821 q^{57} -27.7736i q^{59} +54.1954 q^{61} +5.44096i q^{63} +107.507i q^{67} +60.7101 q^{69} -70.7997i q^{71} -37.4351 q^{73} -23.8356 q^{77} -97.6530i q^{79} -66.2835 q^{81} +126.163i q^{83} -55.1852i q^{87} -133.635 q^{89} -62.0352i q^{91} -72.7645 q^{93} +6.40900 q^{97} +8.74960i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{9}+O(q^{10}) $ 6 * q - 18 * q^9	$ 6 q - 18 q^{9} + 80 q^{17} - 8 q^{21} + 44 q^{29} + 144 q^{33} - 208 q^{37} - 68 q^{41} + 62 q^{49} - 64 q^{53} + 400 q^{57} + 100 q^{61} - 184 q^{69} + 80 q^{73} - 400 q^{77} + 238 q^{81} - 76 q^{89} - 320 q^{93} + 208 q^{97}+O(q^{100}) $ 6 * q - 18 * q^9 + 80 * q^17 - 8 * q^21 + 44 * q^29 + 144 * q^33 - 208 * q^37 - 68 * q^41 + 62 * q^49 - 64 * q^53 + 400 * q^57 + 100 * q^61 - 184 * q^69 + 80 * q^73 - 400 * q^77 + 238 * q^81 - 76 * q^89 - 320 * q^93 + 208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.75441i	0.918135i	0.888401	+	0.459068i	$0.151816\pi$
$3$	2.75441i	0.918135i	−0.888401	+	0.459068i	$0.848184\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.84997i	0.549995i	0.961445	+	0.274998i	$0.0886771\pi$
$7$	3.84997i	0.549995i	−0.961445	+	0.274998i	$0.911323\pi$
$8$	0	0
$9$	1.41325	0.157028
$10$	0	0
$11$	6.19112i	0.562829i	0.959586	+	0.281415i	$0.0908037\pi$
$11$	6.19112i	0.562829i	−0.959586	+	0.281415i	$0.909196\pi$
$12$	0	0
$13$	−16.1132	−1.23948	−0.619738	−	0.784809i	$-0.712761\pi$
$13$	−16.1132	−1.23948	−0.619738	+	0.784809i	$0.712761\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.20875	−0.306397	−0.153198	−	0.988195i	$-0.548957\pi$
$17$	−5.20875	−0.306397	−0.153198	+	0.988195i	$0.548957\pi$
$18$	0	0
$19$	− 36.2264i	− 1.90665i	−0.301944	−	0.953326i	$-0.597636\pi$
$19$	− 36.2264i	− 1.90665i	0.301944	−	0.953326i	$-0.402364\pi$
$20$	0	0
$21$	−10.6044	−0.504970
$22$	0	0
$23$	− 22.0411i	− 0.958308i	−0.877731	−	0.479154i	$-0.840943\pi$
$23$	− 22.0411i	− 0.958308i	0.877731	−	0.479154i	$-0.159057\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	28.6823i	1.06231i
$28$	0	0
$29$	−20.0352	−0.690871	−0.345435	−	0.938443i	$-0.612269\pi$
$29$	−20.0352	−0.690871	−0.345435	+	0.938443i	$0.612269\pi$
$30$	0	0
$31$	26.4175i	0.852177i	0.904681	+	0.426089i	$0.140109\pi$
$31$	26.4175i	0.852177i	−0.904681	+	0.426089i	$0.859891\pi$
$32$	0	0
$33$	−17.0529	−0.516754
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−69.3219	−1.87357	−0.936783	−	0.349911i	$-0.886212\pi$
$37$	−69.3219	−1.87357	−0.936783	+	0.349911i	$0.886212\pi$
$38$	0	0
$39$	− 44.3822i	− 1.13801i
$40$	0	0
$41$	11.6220	0.283463	0.141732	−	0.989905i	$-0.454733\pi$
$41$	11.6220	0.283463	0.141732	+	0.989905i	$0.454733\pi$
$42$	0	0
$43$	− 25.8542i	− 0.601261i	−0.953741	−	0.300630i	$-0.902803\pi$
$43$	− 25.8542i	− 0.601261i	0.953741	−	0.300630i	$-0.0971971\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 66.1853i	− 1.40820i	−0.710102	−	0.704099i	$-0.751351\pi$
$47$	− 66.1853i	− 1.40820i	0.710102	−	0.704099i	$-0.248649\pi$
$48$	0	0
$49$	34.1777	0.697505
$50$	0	0
$51$	− 14.3470i	− 0.281314i
$52$	0	0
$53$	−39.5751	−0.746699	−0.373350	−	0.927691i	$-0.621791\pi$
$53$	−39.5751	−0.746699	−0.373350	+	0.927691i	$0.621791\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	99.7821	1.75056
$58$	0	0
$59$	− 27.7736i	− 0.470739i	−0.971906	−	0.235370i	$-0.924370\pi$
$59$	− 27.7736i	− 0.470739i	0.971906	−	0.235370i	$-0.0756301\pi$
$60$	0	0
$61$	54.1954	0.888449	0.444224	−	0.895916i	$-0.353479\pi$
$61$	54.1954	0.888449	0.444224	+	0.895916i	$0.353479\pi$
$62$	0	0
$63$	5.44096i	0.0863645i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	107.507i	1.60459i	0.596931	+	0.802293i	$0.296387\pi$
$67$	107.507i	1.60459i	−0.596931	+	0.802293i	$0.703613\pi$
$68$	0	0
$69$	60.7101	0.879857
$70$	0	0
$71$	− 70.7997i	− 0.997179i	−0.866838	−	0.498590i	$-0.833851\pi$
$71$	− 70.7997i	− 0.997179i	0.866838	−	0.498590i	$-0.166149\pi$
$72$	0	0
$73$	−37.4351	−0.512810	−0.256405	−	0.966569i	$-0.582538\pi$
$73$	−37.4351	−0.512810	−0.256405	+	0.966569i	$0.582538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−23.8356	−0.309554
$78$	0	0
$79$	− 97.6530i	− 1.23611i	−0.786133	−	0.618057i	$-0.787920\pi$
$79$	− 97.6530i	− 1.23611i	0.786133	−	0.618057i	$-0.212080\pi$
$80$	0	0
$81$	−66.2835	−0.818315
$82$	0	0
$83$	126.163i	1.52003i	0.649904	+	0.760017i	$0.274809\pi$
$83$	126.163i	1.52003i	−0.649904	+	0.760017i	$0.725191\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 55.1852i	− 0.634313i
$88$	0	0
$89$	−133.635	−1.50151	−0.750757	−	0.660579i	$-0.770311\pi$
$89$	−133.635	−1.50151	−0.750757	+	0.660579i	$0.770311\pi$
$90$	0	0
$91$	− 62.0352i	− 0.681706i
$92$	0	0
$93$	−72.7645	−0.782414
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.40900	0.0660722	0.0330361	−	0.999454i	$-0.489482\pi$
$97$	6.40900	0.0660722	0.0330361	+	0.999454i	$0.489482\pi$
$98$	0	0
$99$	8.74960i	0.0883798i
$100$	0	0
$101$	−121.564	−1.20361	−0.601803	−	0.798644i	$-0.705551\pi$
$101$	−121.564	−1.20361	−0.601803	+	0.798644i	$0.705551\pi$
$102$	0	0
$103$	9.95891i	0.0966884i	0.998831	+	0.0483442i	$0.0153944\pi$
$103$	9.95891i	0.0966884i	−0.998831	+	0.0483442i	$0.984606\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 134.842i	− 1.26020i	−0.776512	−	0.630102i	$-0.783013\pi$
$107$	− 134.842i	− 1.26020i	0.776512	−	0.630102i	$-0.216987\pi$
$108$	0	0
$109$	−28.2306	−0.258996	−0.129498	−	0.991580i	$-0.541337\pi$
$109$	−28.2306	−0.258996	−0.129498	+	0.991580i	$0.541337\pi$
$110$	0	0
$111$	− 190.941i	− 1.72019i
$112$	0	0
$113$	190.052	1.68188	0.840939	−	0.541130i	$-0.182003\pi$
$113$	190.052	1.68188	0.840939	+	0.541130i	$0.182003\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−22.7719	−0.194632
$118$	0	0
$119$	− 20.0535i	− 0.168517i
$120$	0	0
$121$	82.6700	0.683223
$122$	0	0
$123$	32.0117i	0.260258i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 60.0646i	− 0.472950i	−0.971638	−	0.236475i	$-0.924008\pi$
$127$	− 60.0646i	− 0.472950i	0.971638	−	0.236475i	$-0.0759921\pi$
$128$	0	0
$129$	71.2130	0.552039
$130$	0	0
$131$	− 111.985i	− 0.854848i	−0.904051	−	0.427424i	$-0.859421\pi$
$131$	− 111.985i	− 0.854848i	0.904051	−	0.427424i	$-0.140579\pi$
$132$	0	0
$133$	139.470	1.04865
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−42.6439	−0.311269	−0.155635	−	0.987815i	$-0.549742\pi$
$137$	−42.6439	−0.311269	−0.155635	+	0.987815i	$0.549742\pi$
$138$	0	0
$139$	− 222.332i	− 1.59951i	−0.600325	−	0.799756i	$-0.704962\pi$
$139$	− 222.332i	− 1.59951i	0.600325	−	0.799756i	$-0.295038\pi$
$140$	0	0
$141$	182.301	1.29292
$142$	0	0
$143$	− 99.7587i	− 0.697614i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	94.1394i	0.640404i
$148$	0	0
$149$	20.0981	0.134887	0.0674434	−	0.997723i	$-0.478516\pi$
$149$	20.0981	0.134887	0.0674434	+	0.997723i	$0.478516\pi$
$150$	0	0
$151$	86.0522i	0.569882i	0.958545	+	0.284941i	$0.0919741\pi$
$151$	86.0522i	0.569882i	−0.958545	+	0.284941i	$0.908026\pi$
$152$	0	0
$153$	−7.36126	−0.0481128
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.0342	0.102129	0.0510643	−	0.998695i	$-0.483739\pi$
$157$	16.0342	0.102129	0.0510643	+	0.998695i	$0.483739\pi$
$158$	0	0
$159$	− 109.006i	− 0.685571i
$160$	0	0
$161$	84.8575	0.527065
$162$	0	0
$163$	− 179.157i	− 1.09912i	−0.835454	−	0.549561i	$-0.814795\pi$
$163$	− 179.157i	− 1.09912i	0.835454	−	0.549561i	$-0.185205\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	137.800i	0.825149i	0.910924	+	0.412574i	$0.135370\pi$
$167$	137.800i	0.825149i	−0.910924	+	0.412574i	$0.864630\pi$
$168$	0	0
$169$	90.6347	0.536300
$170$	0	0
$171$	− 51.1969i	− 0.299397i
$172$	0	0
$173$	62.8895	0.363523	0.181762	−	0.983343i	$-0.441820\pi$
$173$	62.8895	0.363523	0.181762	+	0.983343i	$0.441820\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	76.4998	0.432203
$178$	0	0
$179$	− 238.020i	− 1.32972i	−0.746967	−	0.664861i	$-0.768491\pi$
$179$	− 238.020i	− 1.32972i	0.746967	−	0.664861i	$-0.231509\pi$
$180$	0	0
$181$	−186.718	−1.03159	−0.515795	−	0.856712i	$-0.672503\pi$
$181$	−186.718	−1.03159	−0.515795	+	0.856712i	$0.672503\pi$
$182$	0	0
$183$	149.276i	0.815716i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 32.2480i	− 0.172449i
$188$	0	0
$189$	−110.426	−0.584264
$190$	0	0
$191$	123.447i	0.646319i	0.946344	+	0.323160i	$0.104745\pi$
$191$	123.447i	0.646319i	−0.946344	+	0.323160i	$0.895255\pi$
$192$	0	0
$193$	−162.355	−0.841220	−0.420610	−	0.907242i	$-0.638184\pi$
$193$	−162.355	−0.841220	−0.420610	+	0.907242i	$0.638184\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	113.540	0.576344	0.288172	−	0.957579i	$-0.406952\pi$
$197$	113.540	0.576344	0.288172	+	0.957579i	$0.406952\pi$
$198$	0	0
$199$	− 325.928i	− 1.63783i	−0.573915	−	0.818915i	$-0.694576\pi$
$199$	− 325.928i	− 1.63783i	0.573915	−	0.818915i	$-0.305424\pi$
$200$	0	0
$201$	−296.118	−1.47323
$202$	0	0
$203$	− 77.1351i	− 0.379976i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 31.1496i	− 0.150481i
$208$	0	0
$209$	224.282	1.07312
$210$	0	0
$211$	130.731i	0.619580i	0.950805	+	0.309790i	$0.100259\pi$
$211$	130.731i	0.619580i	−0.950805	+	0.309790i	$0.899741\pi$
$212$	0	0
$213$	195.011	0.915546
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−101.707	−0.468694
$218$	0	0
$219$	− 103.112i	− 0.470829i
$220$	0	0
$221$	83.9295	0.379772
$222$	0	0
$223$	93.3889i	0.418784i	0.977832	+	0.209392i	$0.0671485\pi$
$223$	93.3889i	0.418784i	−0.977832	+	0.209392i	$0.932851\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 14.9957i	− 0.0660602i	−0.999454	−	0.0330301i	$-0.989484\pi$
$227$	− 14.9957i	− 0.0660602i	0.999454	−	0.0330301i	$-0.0105157\pi$
$228$	0	0
$229$	−144.106	−0.629283	−0.314641	−	0.949211i	$-0.601884\pi$
$229$	−144.106	−0.629283	−0.314641	+	0.949211i	$0.601884\pi$
$230$	0	0
$231$	− 65.6530i	− 0.284212i
$232$	0	0
$233$	126.528	0.543040	0.271520	−	0.962433i	$-0.412474\pi$
$233$	126.528	0.543040	0.271520	+	0.962433i	$0.412474\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	268.976	1.13492
$238$	0	0
$239$	− 1.65300i	− 0.00691630i	−0.999994	−	0.00345815i	$-0.998899\pi$
$239$	− 1.65300i	− 0.00691630i	0.999994	−	0.00345815i	$-0.00110077\pi$
$240$	0	0
$241$	206.928	0.858622	0.429311	−	0.903157i	$-0.358756\pi$
$241$	206.928	0.858622	0.429311	+	0.903157i	$0.358756\pi$
$242$	0	0
$243$	75.5692i	0.310984i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	583.722i	2.36325i
$248$	0	0
$249$	−347.503	−1.39560
$250$	0	0
$251$	74.1206i	0.295301i	0.989040	+	0.147651i	$0.0471711\pi$
$251$	74.1206i	0.295301i	−0.989040	+	0.147651i	$0.952829\pi$
$252$	0	0
$253$	136.459	0.539364
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−274.682	−1.06880	−0.534402	−	0.845231i	$-0.679463\pi$
$257$	−274.682	−1.06880	−0.534402	+	0.845231i	$0.679463\pi$
$258$	0	0
$259$	− 266.887i	− 1.03045i
$260$	0	0
$261$	−28.3148	−0.108486
$262$	0	0
$263$	− 75.5382i	− 0.287218i	−0.989635	−	0.143609i	$-0.954129\pi$
$263$	− 75.5382i	− 0.287218i	0.989635	−	0.143609i	$-0.0458707\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 368.084i	− 1.37859i
$268$	0	0
$269$	314.087	1.16761	0.583804	−	0.811895i	$-0.301564\pi$
$269$	314.087	1.16761	0.583804	+	0.811895i	$0.301564\pi$
$270$	0	0
$271$	128.158i	0.472908i	0.971643	+	0.236454i	$0.0759852\pi$
$271$	128.158i	0.472908i	−0.971643	+	0.236454i	$0.924015\pi$
$272$	0	0
$273$	170.870	0.625898
$274$	0	0
$275$	0	0
$276$	0	0
$277$	242.118	0.874073	0.437037	−	0.899444i	$-0.356028\pi$
$277$	242.118	0.874073	0.437037	+	0.899444i	$0.356028\pi$
$278$	0	0
$279$	37.3345i	0.133815i
$280$	0	0
$281$	28.8562	0.102691	0.0513456	−	0.998681i	$-0.483649\pi$
$281$	28.8562	0.102691	0.0513456	+	0.998681i	$0.483649\pi$
$282$	0	0
$283$	− 269.993i	− 0.954039i	−0.878893	−	0.477020i	$-0.841717\pi$
$283$	− 269.993i	− 0.954039i	0.878893	−	0.477020i	$-0.158283\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	44.7443i	0.155904i
$288$	0	0
$289$	−261.869	−0.906121
$290$	0	0
$291$	17.6530i	0.0606632i
$292$	0	0
$293$	−353.448	−1.20631	−0.603154	−	0.797625i	$-0.706089\pi$
$293$	−353.448	−1.20631	−0.603154	+	0.797625i	$0.706089\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−177.576	−0.597898
$298$	0	0
$299$	355.152i	1.18780i
$300$	0	0
$301$	99.5379	0.330691
$302$	0	0
$303$	− 334.837i	− 1.10507i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 260.946i	− 0.849985i	−0.905197	−	0.424993i	$-0.860277\pi$
$307$	− 260.946i	− 0.849985i	0.905197	−	0.424993i	$-0.139723\pi$
$308$	0	0
$309$	−27.4309	−0.0887730
$310$	0	0
$311$	− 141.570i	− 0.455208i	−0.973754	−	0.227604i	$-0.926911\pi$
$311$	− 141.570i	− 0.455208i	0.973754	−	0.227604i	$-0.0730892\pi$
$312$	0	0
$313$	−365.950	−1.16917	−0.584584	−	0.811333i	$-0.698742\pi$
$313$	−365.950	−1.16917	−0.584584	+	0.811333i	$0.698742\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.22805	−0.0291106	−0.0145553	−	0.999894i	$-0.504633\pi$
$317$	−9.22805	−0.0291106	−0.0145553	+	0.999894i	$0.504633\pi$
$318$	0	0
$319$	− 124.041i	− 0.388842i
$320$	0	0
$321$	371.409	1.15704
$322$	0	0
$323$	188.694i	0.584192i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 77.7586i	− 0.237794i
$328$	0	0
$329$	254.811	0.774502
$330$	0	0
$331$	− 53.3799i	− 0.161268i	−0.996744	−	0.0806342i	$-0.974305\pi$
$331$	− 53.3799i	− 0.161268i	0.996744	−	0.0806342i	$-0.0256946\pi$
$332$	0	0
$333$	−97.9692	−0.294202
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−350.458	−1.03994	−0.519968	−	0.854186i	$-0.674056\pi$
$337$	−350.458	−1.03994	−0.519968	+	0.854186i	$0.674056\pi$
$338$	0	0
$339$	523.481i	1.54419i
$340$	0	0
$341$	−163.554	−0.479630
$342$	0	0
$343$	320.232i	0.933620i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	70.8302i	0.204122i	0.994778	+	0.102061i	$0.0325436\pi$
$347$	70.8302i	0.204122i	−0.994778	+	0.102061i	$0.967456\pi$
$348$	0	0
$349$	373.045	1.06890	0.534449	−	0.845201i	$-0.320519\pi$
$349$	373.045	1.06890	0.534449	+	0.845201i	$0.320519\pi$
$350$	0	0
$351$	− 462.163i	− 1.31670i
$352$	0	0
$353$	−543.568	−1.53985	−0.769927	−	0.638132i	$-0.779707\pi$
$353$	−543.568	−1.53985	−0.769927	+	0.638132i	$0.779707\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	55.2355	0.154721
$358$	0	0
$359$	500.805i	1.39500i	0.716585	+	0.697500i	$0.245704\pi$
$359$	500.805i	1.39500i	−0.716585	+	0.697500i	$0.754296\pi$
$360$	0	0
$361$	−951.350	−2.63532
$362$	0	0
$363$	227.707i	0.627291i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 142.499i	− 0.388281i	−0.980974	−	0.194140i	$-0.937808\pi$
$367$	− 142.499i	− 0.388281i	0.980974	−	0.194140i	$-0.0621918\pi$
$368$	0	0
$369$	16.4248	0.0445116
$370$	0	0
$371$	− 152.363i	− 0.410681i
$372$	0	0
$373$	−160.000	−0.428954	−0.214477	−	0.976729i	$-0.568805\pi$
$373$	−160.000	−0.428954	−0.214477	+	0.976729i	$0.568805\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	322.832	0.856317
$378$	0	0
$379$	192.796i	0.508698i	0.967113	+	0.254349i	$0.0818611\pi$
$379$	192.796i	0.508698i	−0.967113	+	0.254349i	$0.918139\pi$
$380$	0	0
$381$	165.442	0.434232
$382$	0	0
$383$	605.286i	1.58038i	0.612861	+	0.790191i	$0.290019\pi$
$383$	605.286i	1.58038i	−0.612861	+	0.790191i	$0.709981\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 36.5384i	− 0.0944146i
$388$	0	0
$389$	−522.159	−1.34231	−0.671155	−	0.741317i	$-0.734202\pi$
$389$	−522.159	−1.34231	−0.671155	+	0.741317i	$0.734202\pi$
$390$	0	0
$391$	114.806i	0.293623i
$392$	0	0
$393$	308.452	0.784866
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−357.537	−0.900598	−0.450299	−	0.892878i	$-0.648683\pi$
$397$	−357.537	−0.900598	−0.450299	+	0.892878i	$0.648683\pi$
$398$	0	0
$399$	384.158i	0.962802i
$400$	0	0
$401$	262.506	0.654629	0.327315	−	0.944915i	$-0.393856\pi$
$401$	262.506	0.654629	0.327315	+	0.944915i	$0.393856\pi$
$402$	0	0
$403$	− 425.670i	− 1.05625i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 429.181i	− 1.05450i
$408$	0	0
$409$	63.2015	0.154527	0.0772634	−	0.997011i	$-0.475382\pi$
$409$	63.2015	0.154527	0.0772634	+	0.997011i	$0.475382\pi$
$410$	0	0
$411$	− 117.459i	− 0.285787i
$412$	0	0
$413$	106.928	0.258905
$414$	0	0
$415$	0	0
$416$	0	0
$417$	612.393	1.46857
$418$	0	0
$419$	673.390i	1.60714i	0.595213	+	0.803568i	$0.297068\pi$
$419$	673.390i	1.60714i	−0.595213	+	0.803568i	$0.702932\pi$
$420$	0	0
$421$	−84.6877	−0.201158	−0.100579	−	0.994929i	$-0.532070\pi$
$421$	−84.6877	−0.201158	−0.100579	+	0.994929i	$0.532070\pi$
$422$	0	0
$423$	− 93.5363i	− 0.221126i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	208.650i	0.488643i
$428$	0	0
$429$	274.776	0.640504
$430$	0	0
$431$	672.158i	1.55953i	0.626072	+	0.779766i	$0.284662\pi$
$431$	672.158i	1.55953i	−0.626072	+	0.779766i	$0.715338\pi$
$432$	0	0
$433$	−562.185	−1.29835	−0.649174	−	0.760640i	$-0.724885\pi$
$433$	−562.185	−1.29835	−0.649174	+	0.760640i	$0.724885\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−798.469	−1.82716
$438$	0	0
$439$	384.842i	0.876633i	0.898821	+	0.438316i	$0.144425\pi$
$439$	384.842i	0.876633i	−0.898821	+	0.438316i	$0.855575\pi$
$440$	0	0
$441$	48.3017	0.109528
$442$	0	0
$443$	461.625i	1.04204i	0.853544	+	0.521021i	$0.174449\pi$
$443$	461.625i	1.04204i	−0.853544	+	0.521021i	$0.825551\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	55.3584i	0.123844i
$448$	0	0
$449$	−48.7390	−0.108550	−0.0542750	−	0.998526i	$-0.517285\pi$
$449$	−48.7390	−0.108550	−0.0542750	+	0.998526i	$0.517285\pi$
$450$	0	0
$451$	71.9532i	0.159542i
$452$	0	0
$453$	−237.023	−0.523229
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−762.588	−1.66868	−0.834341	−	0.551248i	$-0.814152\pi$
$457$	−762.588	−1.66868	−0.834341	+	0.551248i	$0.814152\pi$
$458$	0	0
$459$	− 149.399i	− 0.325488i
$460$	0	0
$461$	406.436	0.881639	0.440820	−	0.897596i	$-0.354688\pi$
$461$	406.436	0.881639	0.440820	+	0.897596i	$0.354688\pi$
$462$	0	0
$463$	− 260.743i	− 0.563159i	−0.959538	−	0.281580i	$-0.909142\pi$
$463$	− 260.743i	− 0.563159i	0.959538	−	0.281580i	$-0.0908583\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	594.738i	1.27353i	0.771058	+	0.636765i	$0.219728\pi$
$467$	594.738i	1.27353i	−0.771058	+	0.636765i	$0.780272\pi$
$468$	0	0
$469$	−413.899	−0.882515
$470$	0	0
$471$	44.1647i	0.0937679i
$472$	0	0
$473$	160.067	0.338407
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−55.9294	−0.117252
$478$	0	0
$479$	534.894i	1.11669i	0.829609	+	0.558344i	$0.188563\pi$
$479$	534.894i	1.11669i	−0.829609	+	0.558344i	$0.811437\pi$
$480$	0	0
$481$	1117.00	2.32224
$482$	0	0
$483$	233.732i	0.483917i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 264.298i	− 0.542706i	−0.962480	−	0.271353i	$-0.912529\pi$
$487$	− 264.298i	− 0.542706i	0.962480	−	0.271353i	$-0.0874711\pi$
$488$	0	0
$489$	493.471	1.00914
$490$	0	0
$491$	539.150i	1.09807i	0.835801	+	0.549033i	$0.185004\pi$
$491$	539.150i	1.09807i	−0.835801	+	0.549033i	$0.814996\pi$
$492$	0	0
$493$	104.359	0.211681
$494$	0	0
$495$	0	0
$496$	0	0
$497$	272.577	0.548444
$498$	0	0
$499$	− 138.218i	− 0.276991i	−0.990363	−	0.138495i	$-0.955773\pi$
$499$	− 138.218i	− 0.276991i	0.990363	−	0.138495i	$-0.0442266\pi$
$500$	0	0
$501$	−379.557	−0.757598
$502$	0	0
$503$	− 389.170i	− 0.773697i	−0.922143	−	0.386848i	$-0.873564\pi$
$503$	− 389.170i	− 0.773697i	0.922143	−	0.386848i	$-0.126436\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	249.645i	0.492396i
$508$	0	0
$509$	−468.599	−0.920627	−0.460314	−	0.887756i	$-0.652263\pi$
$509$	−468.599	−0.920627	−0.460314	+	0.887756i	$0.652263\pi$
$510$	0	0
$511$	− 144.124i	− 0.282043i
$512$	0	0
$513$	1039.06	2.02545
$514$	0	0
$515$	0	0
$516$	0	0
$517$	409.761	0.792575
$518$	0	0
$519$	173.223i	0.333764i
$520$	0	0
$521$	−931.151	−1.78724	−0.893619	−	0.448826i	$-0.851842\pi$
$521$	−931.151	−1.78724	−0.893619	+	0.448826i	$0.851842\pi$
$522$	0	0
$523$	− 227.656i	− 0.435289i	−0.976028	−	0.217645i	$-0.930163\pi$
$523$	− 227.656i	− 0.435289i	0.976028	−	0.217645i	$-0.0698374\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 137.602i	− 0.261104i
$528$	0	0
$529$	43.1902	0.0816451
$530$	0	0
$531$	− 39.2511i	− 0.0739191i
$532$	0	0
$533$	−187.267	−0.351346
$534$	0	0
$535$	0	0
$536$	0	0
$537$	655.605	1.22087
$538$	0	0
$539$	211.599i	0.392576i
$540$	0	0
$541$	−388.174	−0.717511	−0.358756	−	0.933431i	$-0.616799\pi$
$541$	−388.174	−0.717511	−0.358756	+	0.933431i	$0.616799\pi$
$542$	0	0
$543$	− 514.296i	− 0.947139i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 473.059i	− 0.864824i	−0.901676	−	0.432412i	$-0.857663\pi$
$547$	− 473.059i	− 0.864824i	0.901676	−	0.432412i	$-0.142337\pi$
$548$	0	0
$549$	76.5916	0.139511
$550$	0	0
$551$	725.804i	1.31725i
$552$	0	0
$553$	375.961	0.679857
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−419.101	−0.752426	−0.376213	−	0.926533i	$-0.622774\pi$
$557$	−419.101	−0.752426	−0.376213	+	0.926533i	$0.622774\pi$
$558$	0	0
$559$	416.594i	0.745248i
$560$	0	0
$561$	88.8241	0.158332
$562$	0	0
$563$	− 145.910i	− 0.259165i	−0.991569	−	0.129582i	$-0.958636\pi$
$563$	− 145.910i	− 0.259165i	0.991569	−	0.129582i	$-0.0413637\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 255.189i	− 0.450069i
$568$	0	0
$569$	−950.513	−1.67050	−0.835249	−	0.549872i	$-0.814677\pi$
$569$	−950.513	−1.67050	−0.835249	+	0.549872i	$0.814677\pi$
$570$	0	0
$571$	404.107i	0.707717i	0.935299	+	0.353859i	$0.115131\pi$
$571$	404.107i	0.707717i	−0.935299	+	0.353859i	$0.884869\pi$
$572$	0	0
$573$	−340.023	−0.593408
$574$	0	0
$575$	0	0
$576$	0	0
$577$	847.944	1.46957	0.734787	−	0.678298i	$-0.237282\pi$
$577$	847.944	1.46957	0.734787	+	0.678298i	$0.237282\pi$
$578$	0	0
$579$	− 447.193i	− 0.772354i
$580$	0	0
$581$	−485.723	−0.836011
$582$	0	0
$583$	− 245.014i	− 0.420264i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	658.243i	1.12137i	0.828030	+	0.560684i	$0.189462\pi$
$587$	658.243i	1.12137i	−0.828030	+	0.560684i	$0.810538\pi$
$588$	0	0
$589$	957.010	1.62480
$590$	0	0
$591$	312.735i	0.529162i
$592$	0	0
$593$	282.430	0.476274	0.238137	−	0.971232i	$-0.423463\pi$
$593$	282.430	0.476274	0.238137	+	0.971232i	$0.423463\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	897.739	1.50375
$598$	0	0
$599$	− 498.597i	− 0.832382i	−0.909277	−	0.416191i	$-0.863365\pi$
$599$	− 498.597i	− 0.832382i	0.909277	−	0.416191i	$-0.136635\pi$
$600$	0	0
$601$	−287.496	−0.478363	−0.239182	−	0.970975i	$-0.576879\pi$
$601$	−287.496	−0.478363	−0.239182	+	0.970975i	$0.576879\pi$
$602$	0	0
$603$	151.934i	0.251964i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	844.260i	1.39087i	0.718587	+	0.695437i	$0.244789\pi$
$607$	844.260i	1.39087i	−0.718587	+	0.695437i	$0.755211\pi$
$608$	0	0
$609$	212.461	0.348869
$610$	0	0
$611$	1066.46i	1.74543i
$612$	0	0
$613$	975.340	1.59109	0.795547	−	0.605892i	$-0.207184\pi$
$613$	975.340	1.59109	0.795547	+	0.605892i	$0.207184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	319.229	0.517389	0.258695	−	0.965959i	$-0.416708\pi$
$617$	319.229	0.517389	0.258695	+	0.965959i	$0.416708\pi$
$618$	0	0
$619$	845.837i	1.36646i	0.730205	+	0.683228i	$0.239425\pi$
$619$	845.837i	1.36646i	−0.730205	+	0.683228i	$0.760575\pi$
$620$	0	0
$621$	632.190	1.01802
$622$	0	0
$623$	− 514.489i	− 0.825826i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	617.764i	0.985269i
$628$	0	0
$629$	361.080	0.574055
$630$	0	0
$631$	− 322.653i	− 0.511335i	−0.966765	−	0.255668i	$-0.917705\pi$
$631$	− 322.653i	− 0.511335i	0.966765	−	0.255668i	$-0.0822953\pi$
$632$	0	0
$633$	−360.087	−0.568858
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−550.712	−0.864541
$638$	0	0
$639$	− 100.058i	− 0.156585i
$640$	0	0
$641$	167.659	0.261558	0.130779	−	0.991412i	$-0.458252\pi$
$641$	167.659	0.261558	0.130779	+	0.991412i	$0.458252\pi$
$642$	0	0
$643$	118.227i	0.183867i	0.995765	+	0.0919335i	$0.0293047\pi$
$643$	118.227i	0.183867i	−0.995765	+	0.0919335i	$0.970695\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 783.464i	− 1.21092i	−0.795876	−	0.605459i	$-0.792989\pi$
$647$	− 783.464i	− 1.21092i	0.795876	−	0.605459i	$-0.207011\pi$
$648$	0	0
$649$	171.950	0.264946
$650$	0	0
$651$	− 280.141i	− 0.430324i
$652$	0	0
$653$	−28.4352	−0.0435455	−0.0217727	−	0.999763i	$-0.506931\pi$
$653$	−28.4352	−0.0435455	−0.0217727	+	0.999763i	$0.506931\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−52.9051	−0.0805253
$658$	0	0
$659$	594.296i	0.901814i	0.892571	+	0.450907i	$0.148899\pi$
$659$	594.296i	0.901814i	−0.892571	+	0.450907i	$0.851101\pi$
$660$	0	0
$661$	495.511	0.749638	0.374819	−	0.927098i	$-0.377705\pi$
$661$	495.511	0.749638	0.374819	+	0.927098i	$0.377705\pi$
$662$	0	0
$663$	231.176i	0.348682i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	441.599i	0.662067i
$668$	0	0
$669$	−257.231	−0.384501
$670$	0	0
$671$	335.530i	0.500045i
$672$	0	0
$673$	168.874	0.250927	0.125464	−	0.992098i	$-0.459958\pi$
$673$	168.874	0.250927	0.125464	+	0.992098i	$0.459958\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−895.899	−1.32334	−0.661668	−	0.749797i	$-0.730151\pi$
$677$	−895.899	−1.32334	−0.661668	+	0.749797i	$0.730151\pi$
$678$	0	0
$679$	24.6745i	0.0363394i
$680$	0	0
$681$	41.3042	0.0606522
$682$	0	0
$683$	− 359.410i	− 0.526223i	−0.964765	−	0.263112i	$-0.915251\pi$
$683$	− 359.410i	− 0.526223i	0.964765	−	0.263112i	$-0.0847487\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 396.926i	− 0.577767i
$688$	0	0
$689$	637.680	0.925516
$690$	0	0
$691$	− 515.701i	− 0.746311i	−0.927769	−	0.373155i	$-0.878276\pi$
$691$	− 515.701i	− 0.746311i	0.927769	−	0.373155i	$-0.121724\pi$
$692$	0	0
$693$	−33.6857	−0.0486085
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−60.5360	−0.0868523
$698$	0	0
$699$	348.510i	0.498584i
$700$	0	0
$701$	1370.37	1.95488	0.977438	−	0.211221i	$-0.0677441\pi$
$701$	1370.37	1.95488	0.977438	+	0.211221i	$0.0677441\pi$
$702$	0	0
$703$	2511.28i	3.57224i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 468.018i	− 0.661978i
$708$	0	0
$709$	−662.128	−0.933890	−0.466945	−	0.884286i	$-0.654645\pi$
$709$	−662.128	−0.933890	−0.466945	+	0.884286i	$0.654645\pi$
$710$	0	0
$711$	− 138.008i	− 0.194104i
$712$	0	0
$713$	582.270	0.816649
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.55302	0.00635010
$718$	0	0
$719$	− 370.003i	− 0.514608i	−0.966331	−	0.257304i	$-0.917166\pi$
$719$	− 370.003i	− 0.514608i	0.966331	−	0.257304i	$-0.0828341\pi$
$720$	0	0
$721$	−38.3415	−0.0531782
$722$	0	0
$723$	569.964i	0.788331i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 607.695i	− 0.835894i	−0.908471	−	0.417947i	$-0.862750\pi$
$727$	− 607.695i	− 0.835894i	0.908471	−	0.417947i	$-0.137250\pi$
$728$	0	0
$729$	−804.700	−1.10384
$730$	0	0
$731$	134.668i	0.184224i
$732$	0	0
$733$	1066.76	1.45533	0.727666	−	0.685931i	$-0.240605\pi$
$733$	1066.76	1.45533	0.727666	+	0.685931i	$0.240605\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−665.591	−0.903108
$738$	0	0
$739$	558.366i	0.755570i	0.925893	+	0.377785i	$0.123314\pi$
$739$	558.366i	0.755570i	−0.925893	+	0.377785i	$0.876686\pi$
$740$	0	0
$741$	−1607.81	−2.16978
$742$	0	0
$743$	− 1112.00i	− 1.49664i	−0.663339	−	0.748319i	$-0.730861\pi$
$743$	− 1112.00i	− 1.49664i	0.663339	−	0.748319i	$-0.269139\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	178.299i	0.238687i
$748$	0	0
$749$	519.137	0.693107
$750$	0	0
$751$	− 1207.34i	− 1.60764i	−0.594873	−	0.803820i	$-0.702798\pi$
$751$	− 1207.34i	− 1.60764i	0.594873	−	0.803820i	$-0.297202\pi$
$752$	0	0
$753$	−204.158	−0.271127
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−87.7776	−0.115955	−0.0579773	−	0.998318i	$-0.518465\pi$
$757$	−87.7776	−0.115955	−0.0579773	+	0.998318i	$0.518465\pi$
$758$	0	0
$759$	375.864i	0.495209i
$760$	0	0
$761$	67.0202	0.0880686	0.0440343	−	0.999030i	$-0.485979\pi$
$761$	67.0202	0.0880686	0.0440343	+	0.999030i	$0.485979\pi$
$762$	0	0
$763$	− 108.687i	− 0.142447i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	447.522i	0.583470i
$768$	0	0
$769$	−342.869	−0.445863	−0.222932	−	0.974834i	$-0.571563\pi$
$769$	−342.869	−0.445863	−0.222932	+	0.974834i	$0.571563\pi$
$770$	0	0
$771$	− 756.587i	− 0.981306i
$772$	0	0
$773$	−244.756	−0.316631	−0.158316	−	0.987389i	$-0.550606\pi$
$773$	−244.756	−0.316631	−0.158316	+	0.987389i	$0.550606\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	735.116	0.946095
$778$	0	0
$779$	− 421.023i	− 0.540466i
$780$	0	0
$781$	438.330	0.561242
$782$	0	0
$783$	− 574.657i	− 0.733917i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1120.38i	− 1.42361i	−0.702376	−	0.711806i	$-0.747877\pi$
$787$	− 1120.38i	− 1.42361i	0.702376	−	0.711806i	$-0.252123\pi$
$788$	0	0
$789$	208.063	0.263705
$790$	0	0
$791$	731.695i	0.925025i
$792$	0	0
$793$	−873.260	−1.10121
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1344.51	1.68696	0.843480	−	0.537161i	$-0.180503\pi$
$797$	1344.51	1.68696	0.843480	+	0.537161i	$0.180503\pi$
$798$	0	0
$799$	344.742i	0.431467i
$800$	0	0
$801$	−188.859	−0.235779
$802$	0	0
$803$	− 231.765i	− 0.288624i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	865.122i	1.07202i
$808$	0	0
$809$	−345.363	−0.426901	−0.213451	−	0.976954i	$-0.568470\pi$
$809$	−345.363	−0.426901	−0.213451	+	0.976954i	$0.568470\pi$
$810$	0	0
$811$	− 1275.44i	− 1.57268i	−0.617797	−	0.786338i	$-0.711975\pi$
$811$	− 1275.44i	− 1.57268i	0.617797	−	0.786338i	$-0.288025\pi$
$812$	0	0
$813$	−352.999	−0.434193
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−936.604	−1.14639
$818$	0	0
$819$	− 87.6713i	− 0.107047i
$820$	0	0
$821$	−1257.99	−1.53226	−0.766131	−	0.642684i	$-0.777821\pi$
$821$	−1257.99	−1.53226	−0.766131	+	0.642684i	$0.777821\pi$
$822$	0	0
$823$	206.093i	0.250417i	0.992130	+	0.125208i	$0.0399600\pi$
$823$	206.093i	0.250417i	−0.992130	+	0.125208i	$0.960040\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 615.606i	− 0.744384i	−0.928156	−	0.372192i	$-0.878606\pi$
$827$	− 615.606i	− 0.744384i	0.928156	−	0.372192i	$-0.121394\pi$
$828$	0	0
$829$	−1214.12	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$829$	−1214.12	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$830$	0	0
$831$	666.892i	0.802518i
$832$	0	0
$833$	−178.023	−0.213713
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−757.715	−0.905275
$838$	0	0
$839$	1338.99i	1.59593i	0.602701	+	0.797967i	$0.294091\pi$
$839$	1338.99i	1.59593i	−0.602701	+	0.797967i	$0.705909\pi$
$840$	0	0
$841$	−439.589	−0.522698
$842$	0	0
$843$	79.4817i	0.0942844i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	318.277i	0.375769i
$848$	0	0
$849$	743.671	0.875937
$850$	0	0
$851$	1527.93i	1.79545i
$852$	0	0
$853$	−112.066	−0.131379	−0.0656893	−	0.997840i	$-0.520925\pi$
$853$	−112.066	−0.131379	−0.0656893	+	0.997840i	$0.520925\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1089.15	−1.27089	−0.635443	−	0.772148i	$-0.719183\pi$
$857$	−1089.15	−1.27089	−0.635443	+	0.772148i	$0.719183\pi$
$858$	0	0
$859$	− 719.610i	− 0.837729i	−0.908049	−	0.418865i	$-0.862428\pi$
$859$	− 719.610i	− 0.837729i	0.908049	−	0.418865i	$-0.137572\pi$
$860$	0	0
$861$	−123.244	−0.143141
$862$	0	0
$863$	114.135i	0.132254i	0.997811	+	0.0661269i	$0.0210642\pi$
$863$	114.135i	0.132254i	−0.997811	+	0.0661269i	$0.978936\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 721.293i	− 0.831942i
$868$	0	0
$869$	604.582	0.695721
$870$	0	0
$871$	− 1732.28i	− 1.98884i
$872$	0	0
$873$	9.05752	0.0103752
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1427.99	1.62827	0.814135	−	0.580675i	$-0.197211\pi$
$877$	1427.99	1.62827	0.814135	+	0.580675i	$0.197211\pi$
$878$	0	0
$879$	− 973.539i	− 1.10755i
$880$	0	0
$881$	364.165	0.413354	0.206677	−	0.978409i	$-0.433735\pi$
$881$	364.165	0.413354	0.206677	+	0.978409i	$0.433735\pi$
$882$	0	0
$883$	− 800.458i	− 0.906521i	−0.891378	−	0.453260i	$-0.850261\pi$
$883$	− 800.458i	− 0.906521i	0.891378	−	0.453260i	$-0.149739\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	591.672i	0.667049i	0.942741	+	0.333524i	$0.108238\pi$
$887$	591.672i	0.667049i	−0.942741	+	0.333524i	$0.891762\pi$
$888$	0	0
$889$	231.247	0.260120
$890$	0	0
$891$	− 410.369i	− 0.460572i
$892$	0	0
$893$	−2397.65	−2.68494
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−978.233	−1.09056
$898$	0	0
$899$	− 529.281i	− 0.588744i
$900$	0	0
$901$	206.136	0.228786
$902$	0	0
$903$	274.168i	0.303619i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 545.657i	− 0.601606i	−0.953686	−	0.300803i	$-0.902745\pi$
$907$	− 545.657i	− 0.601606i	0.953686	−	0.300803i	$-0.0972547\pi$
$908$	0	0
$909$	−171.801	−0.189000
$910$	0	0
$911$	− 929.286i	− 1.02007i	−0.860153	−	0.510036i	$-0.829632\pi$
$911$	− 929.286i	− 1.02007i	0.860153	−	0.510036i	$-0.170368\pi$
$912$	0	0
$913$	−781.089	−0.855520
$914$	0	0
$915$	0	0
$916$	0	0
$917$	431.139	0.470163
$918$	0	0
$919$	− 171.353i	− 0.186456i	−0.995645	−	0.0932278i	$-0.970282\pi$
$919$	− 171.353i	− 0.186456i	0.995645	−	0.0932278i	$-0.0297185\pi$
$920$	0	0
$921$	718.750	0.780402
$922$	0	0
$923$	1140.81i	1.23598i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.0744i	0.0151828i
$928$	0	0
$929$	1178.62	1.26870	0.634350	−	0.773046i	$-0.281268\pi$
$929$	1178.62	1.26870	0.634350	+	0.773046i	$0.281268\pi$
$930$	0	0
$931$	− 1238.14i	− 1.32990i
$932$	0	0
$933$	389.940	0.417943
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−412.624	−0.440367	−0.220183	−	0.975458i	$-0.570666\pi$
$937$	−412.624	−0.440367	−0.220183	+	0.975458i	$0.570666\pi$
$938$	0	0
$939$	− 1007.97i	− 1.07346i
$940$	0	0
$941$	1121.33	1.19163	0.595817	−	0.803120i	$-0.296828\pi$
$941$	1121.33	1.19163	0.595817	+	0.803120i	$0.296828\pi$
$942$	0	0
$943$	− 256.161i	− 0.271645i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1395.22i	− 1.47331i	−0.676271	−	0.736653i	$-0.736405\pi$
$947$	− 1395.22i	− 1.47331i	0.676271	−	0.736653i	$-0.263595\pi$
$948$	0	0
$949$	603.199	0.635615
$950$	0	0
$951$	− 25.4178i	− 0.0267275i
$952$	0	0
$953$	−476.331	−0.499822	−0.249911	−	0.968269i	$-0.580401\pi$
$953$	−476.331	−0.499822	−0.249911	+	0.968269i	$0.580401\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	341.658	0.357010
$958$	0	0
$959$	− 164.178i	− 0.171197i
$960$	0	0
$961$	263.116	0.273794
$962$	0	0
$963$	− 190.565i	− 0.197887i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 140.846i	− 0.145653i	−0.997345	−	0.0728263i	$-0.976798\pi$
$967$	− 140.846i	− 0.145653i	0.997345	−	0.0728263i	$-0.0232019\pi$
$968$	0	0
$969$	−519.740	−0.536367
$970$	0	0
$971$	1144.98i	1.17918i	0.807704	+	0.589588i	$0.200710\pi$
$971$	1144.98i	1.17918i	−0.807704	+	0.589588i	$0.799290\pi$
$972$	0	0
$973$	855.971	0.879724
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.8912	0.0572070	0.0286035	−	0.999591i	$-0.490894\pi$
$977$	55.8912	0.0572070	0.0286035	+	0.999591i	$0.490894\pi$
$978$	0	0
$979$	− 827.349i	− 0.845096i
$980$	0	0
$981$	−39.8969	−0.0406696
$982$	0	0
$983$	1028.74i	1.04653i	0.852169	+	0.523267i	$0.175287\pi$
$983$	1028.74i	1.04653i	−0.852169	+	0.523267i	$0.824713\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	701.853i	0.711098i
$988$	0	0
$989$	−569.855	−0.576193
$990$	0	0
$991$	666.207i	0.672257i	0.941816	+	0.336129i	$0.109118\pi$
$991$	666.207i	0.672257i	−0.941816	+	0.336129i	$0.890882\pi$
$992$	0	0
$993$	147.030	0.148066
$994$	0	0
$995$	0	0
$996$	0	0
$997$	737.006	0.739224	0.369612	−	0.929186i	$-0.379491\pi$
$997$	737.006	0.739224	0.369612	+	0.929186i	$0.379491\pi$
$998$	0	0
$999$	− 1988.31i	− 1.99030i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.b.w.1151.5		6
4.3	odd	2	inner	1600.3.b.w.1151.2		6
5.2	odd	4		320.3.h.f.319.6		6
5.3	odd	4		320.3.h.g.319.1		6
5.4	even	2		1600.3.b.v.1151.2		6
8.3	odd	2		800.3.b.i.351.5		6
8.5	even	2		800.3.b.i.351.2		6
20.3	even	4		320.3.h.f.319.5		6
20.7	even	4		320.3.h.g.319.2		6
20.19	odd	2		1600.3.b.v.1151.5		6
40.3	even	4		160.3.h.b.159.2	yes	6
40.13	odd	4		160.3.h.a.159.6	yes	6
40.19	odd	2		800.3.b.h.351.2		6
40.27	even	4		160.3.h.a.159.5	✓	6
40.29	even	2		800.3.b.h.351.5		6
40.37	odd	4		160.3.h.b.159.1	yes	6
80.3	even	4		1280.3.e.f.639.5		6
80.13	odd	4		1280.3.e.g.639.2		6
80.27	even	4		1280.3.e.g.639.5		6
80.37	odd	4		1280.3.e.f.639.2		6
80.43	even	4		1280.3.e.h.639.2		6
80.53	odd	4		1280.3.e.i.639.5		6
80.67	even	4		1280.3.e.i.639.2		6
80.77	odd	4		1280.3.e.h.639.5		6
120.53	even	4		1440.3.j.a.1279.1		6
120.77	even	4		1440.3.j.b.1279.2		6
120.83	odd	4		1440.3.j.b.1279.1		6
120.107	odd	4		1440.3.j.a.1279.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.h.a.159.5	✓	6	40.27	even	4
160.3.h.a.159.6	yes	6	40.13	odd	4
160.3.h.b.159.1	yes	6	40.37	odd	4
160.3.h.b.159.2	yes	6	40.3	even	4
320.3.h.f.319.5		6	20.3	even	4
320.3.h.f.319.6		6	5.2	odd	4
320.3.h.g.319.1		6	5.3	odd	4
320.3.h.g.319.2		6	20.7	even	4
800.3.b.h.351.2		6	40.19	odd	2
800.3.b.h.351.5		6	40.29	even	2
800.3.b.i.351.2		6	8.5	even	2
800.3.b.i.351.5		6	8.3	odd	2
1280.3.e.f.639.2		6	80.37	odd	4
1280.3.e.f.639.5		6	80.3	even	4
1280.3.e.g.639.2		6	80.13	odd	4
1280.3.e.g.639.5		6	80.27	even	4
1280.3.e.h.639.2		6	80.43	even	4
1280.3.e.h.639.5		6	80.77	odd	4
1280.3.e.i.639.2		6	80.67	even	4
1280.3.e.i.639.5		6	80.53	odd	4
1440.3.j.a.1279.1		6	120.53	even	4
1440.3.j.a.1279.2		6	120.107	odd	4
1440.3.j.b.1279.1		6	120.83	odd	4
1440.3.j.b.1279.2		6	120.77	even	4
1600.3.b.v.1151.2		6	5.4	even	2
1600.3.b.v.1151.5		6	20.19	odd	2
1600.3.b.w.1151.2		6	4.3	odd	2	inner
1600.3.b.w.1151.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.75441i q^{3} +3.84997i q^{7} +1.41325 q^{9} +O(q^{10})\)	\(q+2.75441i q^{3} +3.84997i q^{7} +1.41325 q^{9} +6.19112i q^{11} -16.1132 q^{13} -5.20875 q^{17} -36.2264i q^{19} -10.6044 q^{21} -22.0411i q^{23} +28.6823i q^{27} -20.0352 q^{29} +26.4175i q^{31} -17.0529 q^{33} -69.3219 q^{37} -44.3822i q^{39} +11.6220 q^{41} -25.8542i q^{43} -66.1853i q^{47} +34.1777 q^{49} -14.3470i q^{51} -39.5751 q^{53} +99.7821 q^{57} -27.7736i q^{59} +54.1954 q^{61} +5.44096i q^{63} +107.507i q^{67} +60.7101 q^{69} -70.7997i q^{71} -37.4351 q^{73} -23.8356 q^{77} -97.6530i q^{79} -66.2835 q^{81} +126.163i q^{83} -55.1852i q^{87} -133.635 q^{89} -62.0352i q^{91} -72.7645 q^{93} +6.40900 q^{97} +8.74960i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{9}+O(q^{10}) \) 6 * q - 18 * q^9	\( 6 q - 18 q^{9} + 80 q^{17} - 8 q^{21} + 44 q^{29} + 144 q^{33} - 208 q^{37} - 68 q^{41} + 62 q^{49} - 64 q^{53} + 400 q^{57} + 100 q^{61} - 184 q^{69} + 80 q^{73} - 400 q^{77} + 238 q^{81} - 76 q^{89} - 320 q^{93} + 208 q^{97}+O(q^{100}) \) 6 * q - 18 * q^9 + 80 * q^17 - 8 * q^21 + 44 * q^29 + 144 * q^33 - 208 * q^37 - 68 * q^41 + 62 * q^49 - 64 * q^53 + 400 * q^57 + 100 * q^61 - 184 * q^69 + 80 * q^73 - 400 * q^77 + 238 * q^81 - 76 * q^89 - 320 * q^93 + 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more