LMFDB - Embedded newform 1600.3.g.j.351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(351,1600)]

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, 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("1600.351");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2342560000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20 $ x^8 - 4x^7 + 24x^6 - 58x^5 + 141x^4 - 190x^3 + 186x^2 - 100*x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.2
Root			$0.500000 + 3.27635i$ of defining polynomial
Character	$\chi$	$=$	1600.351
Dual form			1600.3.g.j.351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.55269 q^{3} +10.4162i q^{7} +21.8324 q^{9} +O(q^{10})$	$q-5.55269 q^{3} +10.4162i q^{7} +21.8324 q^{9} -8.79436 q^{11} +2.31102i q^{13} +18.8324 q^{17} +24.5218 q^{19} -57.8380i q^{21} -19.2486i q^{23} -71.2544 q^{27} -4.62204i q^{29} +40.8324i q^{31} +48.8324 q^{33} -48.5939i q^{37} -12.8324i q^{39} +11.1676 q^{41} -37.0076 q^{43} -80.0810i q^{47} -59.4972 q^{49} -104.571 q^{51} +42.1105i q^{53} -136.162 q^{57} +68.9433 q^{59} -80.0487i q^{61} +227.411i q^{63} -40.7301 q^{67} +106.882i q^{69} -28.1620i q^{71} +35.5028 q^{73} -91.6038i q^{77} +29.6648i q^{79} +199.162 q^{81} +95.2952 q^{83} +25.6648i q^{87} -19.6648 q^{89} -24.0721 q^{91} -226.730i q^{93} +50.1620 q^{97} -192.002 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 56 q^{9}+O(q^{10}) $ 8 * q + 56 * q^9	$ 8 q + 56 q^{9} + 32 q^{17} + 272 q^{33} + 208 q^{41} - 120 q^{49} - 496 q^{57} + 640 q^{73} + 1000 q^{81} + 80 q^{89} - 192 q^{97}+O(q^{100}) $ 8 * q + 56 * q^9 + 32 * q^17 + 272 * q^33 + 208 * q^41 - 120 * q^49 - 496 * q^57 + 640 * q^73 + 1000 * q^81 + 80 * q^89 - 192 * 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
$3$	−5.55269	−1.85090	−0.925449	−	0.378873i	$-0.876312\pi$
$3$	−5.55269	−1.85090	−0.925449	+	0.378873i	$0.876312\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.4162i	1.48803i	0.668164	+	0.744014i	$0.267081\pi$
$7$	10.4162i	1.48803i	−0.668164	+	0.744014i	$0.732919\pi$
$8$	0	0
$9$	21.8324	2.42582
$10$	0	0
$11$	−8.79436	−0.799488	−0.399744	−	0.916627i	$-0.630901\pi$
$11$	−8.79436	−0.799488	−0.399744	+	0.916627i	$0.630901\pi$
$12$	0	0
$13$	2.31102i	0.177771i	0.996042	+	0.0888855i	$0.0283305\pi$
$13$	2.31102i	0.177771i	−0.996042	+	0.0888855i	$0.971669\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.8324	1.10779	0.553894	−	0.832587i	$-0.313141\pi$
$17$	18.8324	1.10779	0.553894	+	0.832587i	$0.313141\pi$
$18$	0	0
$19$	24.5218	1.29062	0.645310	−	0.763921i	$-0.276728\pi$
$19$	24.5218	1.29062	0.645310	+	0.763921i	$0.276728\pi$
$20$	0	0
$21$	− 57.8380i	− 2.75419i
$22$	0	0
$23$	− 19.2486i	− 0.836895i	−0.908241	−	0.418448i	$-0.862574\pi$
$23$	− 19.2486i	− 0.836895i	0.908241	−	0.418448i	$-0.137426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−71.2544	−2.63905
$28$	0	0
$29$	− 4.62204i	− 0.159381i	−0.996820	−	0.0796904i	$-0.974607\pi$
$29$	− 4.62204i	− 0.159381i	0.996820	−	0.0796904i	$-0.0253932\pi$
$30$	0	0
$31$	40.8324i	1.31717i	0.752505	+	0.658587i	$0.228846\pi$
$31$	40.8324i	1.31717i	−0.752505	+	0.658587i	$0.771154\pi$
$32$	0	0
$33$	48.8324	1.47977
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 48.5939i	− 1.31335i	−0.754175	−	0.656674i	$-0.771963\pi$
$37$	− 48.5939i	− 1.31335i	0.754175	−	0.656674i	$-0.228037\pi$
$38$	0	0
$39$	− 12.8324i	− 0.329036i
$40$	0	0
$41$	11.1676	0.272381	0.136190	−	0.990683i	$-0.456514\pi$
$41$	11.1676	0.272381	0.136190	+	0.990683i	$0.456514\pi$
$42$	0	0
$43$	−37.0076	−0.860641	−0.430320	−	0.902676i	$-0.641599\pi$
$43$	−37.0076	−0.860641	−0.430320	+	0.902676i	$0.641599\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 80.0810i	− 1.70385i	−0.523663	−	0.851925i	$-0.675435\pi$
$47$	− 80.0810i	− 1.70385i	0.523663	−	0.851925i	$-0.324565\pi$
$48$	0	0
$49$	−59.4972	−1.21423
$50$	0	0
$51$	−104.571	−2.05040
$52$	0	0
$53$	42.1105i	0.794538i	0.917702	+	0.397269i	$0.130042\pi$
$53$	42.1105i	0.794538i	−0.917702	+	0.397269i	$0.869958\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−136.162	−2.38881
$58$	0	0
$59$	68.9433	1.16853	0.584266	−	0.811563i	$-0.301383\pi$
$59$	68.9433	1.16853	0.584266	+	0.811563i	$0.301383\pi$
$60$	0	0
$61$	− 80.0487i	− 1.31227i	−0.754642	−	0.656137i	$-0.772189\pi$
$61$	− 80.0487i	− 1.31227i	0.754642	−	0.656137i	$-0.227811\pi$
$62$	0	0
$63$	227.411i	3.60969i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−40.7301	−0.607913	−0.303956	−	0.952686i	$-0.598308\pi$
$67$	−40.7301	−0.607913	−0.303956	+	0.952686i	$0.598308\pi$
$68$	0	0
$69$	106.882i	1.54901i
$70$	0	0
$71$	− 28.1620i	− 0.396648i	−0.980137	−	0.198324i	$-0.936450\pi$
$71$	− 28.1620i	− 0.396648i	0.980137	−	0.198324i	$-0.0635498\pi$
$72$	0	0
$73$	35.5028	0.486340	0.243170	−	0.969984i	$-0.421813\pi$
$73$	35.5028	0.486340	0.243170	+	0.969984i	$0.421813\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 91.6038i	− 1.18966i
$78$	0	0
$79$	29.6648i	0.375504i	0.982216	+	0.187752i	$0.0601201\pi$
$79$	29.6648i	0.375504i	−0.982216	+	0.187752i	$0.939880\pi$
$80$	0	0
$81$	199.162	2.45879
$82$	0	0
$83$	95.2952	1.14814	0.574068	−	0.818808i	$-0.305365\pi$
$83$	95.2952	1.14814	0.574068	+	0.818808i	$0.305365\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	25.6648i	0.294998i
$88$	0	0
$89$	−19.6648	−0.220953	−0.110476	−	0.993879i	$-0.535238\pi$
$89$	−19.6648	−0.220953	−0.110476	+	0.993879i	$0.535238\pi$
$90$	0	0
$91$	−24.0721	−0.264528
$92$	0	0
$93$	− 226.730i	− 2.43795i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	50.1620	0.517134	0.258567	−	0.965993i	$-0.416750\pi$
$97$	50.1620	0.517134	0.258567	+	0.965993i	$0.416750\pi$
$98$	0	0
$99$	−192.002	−1.93941
$100$	0	0
$101$	40.6989i	0.402960i	0.979493	+	0.201480i	$0.0645751\pi$
$101$	40.6989i	0.402960i	−0.979493	+	0.201480i	$0.935425\pi$
$102$	0	0
$103$	9.42182i	0.0914739i	0.998954	+	0.0457370i	$0.0145636\pi$
$103$	9.42182i	0.0914739i	−0.998954	+	0.0457370i	$0.985436\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	105.501	0.985992	0.492996	−	0.870032i	$-0.335902\pi$
$107$	105.501	0.985992	0.492996	+	0.870032i	$0.335902\pi$
$108$	0	0
$109$	− 132.815i	− 1.21849i	−0.792984	−	0.609243i	$-0.791474\pi$
$109$	− 132.815i	− 1.21849i	0.792984	−	0.609243i	$-0.208526\pi$
$110$	0	0
$111$	269.827i	2.43087i
$112$	0	0
$113$	112.335	0.994117	0.497058	−	0.867717i	$-0.334413\pi$
$113$	112.335	0.994117	0.497058	+	0.867717i	$0.334413\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	50.4552i	0.431241i
$118$	0	0
$119$	196.162i	1.64842i
$120$	0	0
$121$	−43.6592	−0.360820
$122$	0	0
$123$	−62.0103	−0.504149
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.58942i	− 0.0676332i	−0.999428	−	0.0338166i	$-0.989234\pi$
$127$	− 8.58942i	− 0.0676332i	0.999428	−	0.0338166i	$-0.0107662\pi$
$128$	0	0
$129$	205.492	1.59296
$130$	0	0
$131$	25.4836	0.194532	0.0972658	−	0.995258i	$-0.468990\pi$
$131$	25.4836	0.194532	0.0972658	+	0.995258i	$0.468990\pi$
$132$	0	0
$133$	255.424i	1.92048i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	182.324	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$137$	182.324	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$138$	0	0
$139$	47.6320	0.342676	0.171338	−	0.985212i	$-0.445191\pi$
$139$	47.6320	0.342676	0.171338	+	0.985212i	$0.445191\pi$
$140$	0	0
$141$	444.665i	3.15365i
$142$	0	0
$143$	− 20.3240i	− 0.142126i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	330.370	2.24741
$148$	0	0
$149$	270.702i	1.81679i	0.418114	+	0.908395i	$0.362691\pi$
$149$	270.702i	1.81679i	−0.418114	+	0.908395i	$0.637309\pi$
$150$	0	0
$151$	214.497i	1.42051i	0.703944	+	0.710256i	$0.251421\pi$
$151$	214.497i	1.42051i	−0.703944	+	0.710256i	$0.748579\pi$
$152$	0	0
$153$	411.156	2.68730
$154$	0	0
$155$	0	0
$156$	0	0
$157$	250.290i	1.59420i	0.603846	+	0.797101i	$0.293634\pi$
$157$	250.290i	1.59420i	−0.603846	+	0.797101i	$0.706366\pi$
$158$	0	0
$159$	− 233.827i	− 1.47061i
$160$	0	0
$161$	200.497	1.24532
$162$	0	0
$163$	122.190	0.749635	0.374817	−	0.927099i	$-0.377705\pi$
$163$	122.190	0.749635	0.374817	+	0.927099i	$0.377705\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 35.4106i	− 0.212039i	−0.994364	−	0.106020i	$-0.966189\pi$
$167$	− 35.4106i	− 0.212039i	0.994364	−	0.106020i	$-0.0338107\pi$
$168$	0	0
$169$	163.659	0.968397
$170$	0	0
$171$	535.370	3.13082
$172$	0	0
$173$	− 93.9149i	− 0.542860i	−0.962458	−	0.271430i	$-0.912503\pi$
$173$	− 93.9149i	− 0.542860i	0.962458	−	0.271430i	$-0.0874966\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−382.821	−2.16283
$178$	0	0
$179$	−33.7659	−0.188636	−0.0943181	−	0.995542i	$-0.530067\pi$
$179$	−33.7659	−0.188636	−0.0943181	+	0.995542i	$0.530067\pi$
$180$	0	0
$181$	− 46.2204i	− 0.255362i	−0.991815	−	0.127681i	$-0.959247\pi$
$181$	− 46.2204i	− 0.255362i	0.991815	−	0.127681i	$-0.0407533\pi$
$182$	0	0
$183$	444.486i	2.42888i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−165.619	−0.885663
$188$	0	0
$189$	− 742.200i	− 3.92698i
$190$	0	0
$191$	− 266.151i	− 1.39346i	−0.717334	−	0.696730i	$-0.754638\pi$
$191$	− 266.151i	− 1.39346i	0.717334	−	0.696730i	$-0.245362\pi$
$192$	0	0
$193$	105.168	0.544910	0.272455	−	0.962169i	$-0.412164\pi$
$193$	105.168	0.544910	0.272455	+	0.962169i	$0.412164\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	348.439i	1.76873i	0.466799	+	0.884364i	$0.345407\pi$
$197$	348.439i	1.76873i	−0.466799	+	0.884364i	$0.654593\pi$
$198$	0	0
$199$	− 225.318i	− 1.13225i	−0.824318	−	0.566127i	$-0.808441\pi$
$199$	− 225.318i	− 1.13225i	0.824318	−	0.566127i	$-0.191559\pi$
$200$	0	0
$201$	226.162	1.12518
$202$	0	0
$203$	48.1441	0.237163
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 420.243i	− 2.03016i
$208$	0	0
$209$	−215.654	−1.03184
$210$	0	0
$211$	352.999	1.67298	0.836490	−	0.547982i	$-0.184604\pi$
$211$	352.999	1.67298	0.836490	+	0.547982i	$0.184604\pi$
$212$	0	0
$213$	156.375i	0.734154i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−425.318	−1.95999
$218$	0	0
$219$	−197.136	−0.900165
$220$	0	0
$221$	43.5221i	0.196933i
$222$	0	0
$223$	90.2542i	0.404727i	0.979310	+	0.202364i	$0.0648623\pi$
$223$	90.2542i	0.404727i	−0.979310	+	0.202364i	$0.935138\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	45.3522	0.199789	0.0998947	−	0.994998i	$-0.468149\pi$
$227$	45.3522	0.199789	0.0998947	+	0.994998i	$0.468149\pi$
$228$	0	0
$229$	247.142i	1.07922i	0.841915	+	0.539611i	$0.181428\pi$
$229$	247.142i	1.07922i	−0.841915	+	0.539611i	$0.818572\pi$
$230$	0	0
$231$	508.648i	2.20194i
$232$	0	0
$233$	−133.492	−0.572925	−0.286463	−	0.958091i	$-0.592479\pi$
$233$	−133.492	−0.572925	−0.286463	+	0.958091i	$0.592479\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 164.719i	− 0.695019i
$238$	0	0
$239$	− 8.67041i	− 0.0362779i	−0.999835	−	0.0181389i	$-0.994226\pi$
$239$	− 8.67041i	− 0.0362779i	0.999835	−	0.0181389i	$-0.00577412\pi$
$240$	0	0
$241$	−260.486	−1.08085	−0.540427	−	0.841391i	$-0.681737\pi$
$241$	−260.486	−1.08085	−0.540427	+	0.841391i	$0.681737\pi$
$242$	0	0
$243$	−464.596	−1.91192
$244$	0	0
$245$	0	0
$246$	0	0
$247$	56.6704i	0.229435i
$248$	0	0
$249$	−529.145	−2.12508
$250$	0	0
$251$	−412.311	−1.64267	−0.821336	−	0.570444i	$-0.806771\pi$
$251$	−412.311	−1.64267	−0.821336	+	0.570444i	$0.806771\pi$
$252$	0	0
$253$	169.279i	0.669088i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	312.659	1.21657	0.608286	−	0.793718i	$-0.291857\pi$
$257$	312.659	1.21657	0.608286	+	0.793718i	$0.291857\pi$
$258$	0	0
$259$	506.163	1.95430
$260$	0	0
$261$	− 100.910i	− 0.386630i
$262$	0	0
$263$	61.9078i	0.235391i	0.993050	+	0.117695i	$0.0375506\pi$
$263$	61.9078i	0.235391i	−0.993050	+	0.117695i	$0.962449\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	109.193	0.408961
$268$	0	0
$269$	134.614i	0.500423i	0.968191	+	0.250212i	$0.0805002\pi$
$269$	134.614i	0.500423i	−0.968191	+	0.250212i	$0.919500\pi$
$270$	0	0
$271$	− 265.156i	− 0.978437i	−0.872161	−	0.489218i	$-0.837282\pi$
$271$	− 265.156i	− 0.978437i	0.872161	−	0.489218i	$-0.162718\pi$
$272$	0	0
$273$	133.665	0.489615
$274$	0	0
$275$	0	0
$276$	0	0
$277$	174.413i	0.629651i	0.949150	+	0.314825i	$0.101946\pi$
$277$	174.413i	0.629651i	−0.949150	+	0.314825i	$0.898054\pi$
$278$	0	0
$279$	891.469i	3.19523i
$280$	0	0
$281$	476.810	1.69683	0.848416	−	0.529330i	$-0.177557\pi$
$281$	476.810	1.69683	0.848416	+	0.529330i	$0.177557\pi$
$282$	0	0
$283$	−56.4576	−0.199497	−0.0997484	−	0.995013i	$-0.531804\pi$
$283$	−56.4576	−0.199497	−0.0997484	+	0.995013i	$0.531804\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	116.324i	0.405310i
$288$	0	0
$289$	65.6592	0.227194
$290$	0	0
$291$	−278.534	−0.957162
$292$	0	0
$293$	12.5170i	0.0427200i	0.999772	+	0.0213600i	$0.00679961\pi$
$293$	12.5170i	0.0427200i	−0.999772	+	0.0213600i	$0.993200\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	626.637	2.10989
$298$	0	0
$299$	44.4839	0.148776
$300$	0	0
$301$	− 385.478i	− 1.28066i
$302$	0	0
$303$	− 225.989i	− 0.745838i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−460.811	−1.50101	−0.750507	−	0.660863i	$-0.770190\pi$
$307$	−460.811	−1.50101	−0.750507	+	0.660863i	$0.770190\pi$
$308$	0	0
$309$	− 52.3165i	− 0.169309i
$310$	0	0
$311$	− 112.162i	− 0.360649i	−0.983607	−	0.180325i	$-0.942285\pi$
$311$	− 112.162i	− 0.360649i	0.983607	−	0.180325i	$-0.0577149\pi$
$312$	0	0
$313$	−391.989	−1.25236	−0.626180	−	0.779678i	$-0.715383\pi$
$313$	−391.989	−1.25236	−0.626180	+	0.779678i	$0.715383\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 379.894i	− 1.19840i	−0.800598	−	0.599202i	$-0.795485\pi$
$317$	− 379.894i	− 1.19840i	0.800598	−	0.599202i	$-0.204515\pi$
$318$	0	0
$319$	40.6479i	0.127423i
$320$	0	0
$321$	−585.816	−1.82497
$322$	0	0
$323$	461.804	1.42973
$324$	0	0
$325$	0	0
$326$	0	0
$327$	737.480i	2.25529i
$328$	0	0
$329$	834.140	2.53538
$330$	0	0
$331$	−268.778	−0.812018	−0.406009	−	0.913869i	$-0.633080\pi$
$331$	−268.778	−0.812018	−0.406009	+	0.913869i	$0.633080\pi$
$332$	0	0
$333$	− 1060.92i	− 3.18595i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−149.330	−0.443115	−0.221557	−	0.975147i	$-0.571114\pi$
$337$	−149.330	−0.443115	−0.221557	+	0.975147i	$0.571114\pi$
$338$	0	0
$339$	−623.763	−1.84001
$340$	0	0
$341$	− 359.095i	− 1.05306i
$342$	0	0
$343$	− 109.341i	− 0.318778i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−106.525	−0.306990	−0.153495	−	0.988149i	$-0.549053\pi$
$347$	−106.525	−0.306990	−0.153495	+	0.988149i	$0.549053\pi$
$348$	0	0
$349$	352.674i	1.01053i	0.862965	+	0.505264i	$0.168605\pi$
$349$	352.674i	1.01053i	−0.862965	+	0.505264i	$0.831395\pi$
$350$	0	0
$351$	− 164.670i	− 0.469146i
$352$	0	0
$353$	−400.335	−1.13409	−0.567047	−	0.823685i	$-0.691914\pi$
$353$	−400.335	−1.13409	−0.567047	+	0.823685i	$0.691914\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1089.23i	− 3.05106i
$358$	0	0
$359$	− 93.6423i	− 0.260842i	−0.991459	−	0.130421i	$-0.958367\pi$
$359$	− 93.6423i	− 0.260842i	0.991459	−	0.130421i	$-0.0416329\pi$
$360$	0	0
$361$	240.318	0.665702
$362$	0	0
$363$	242.426	0.667840
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 157.260i	− 0.428501i	−0.976779	−	0.214250i	$-0.931269\pi$
$367$	− 157.260i	− 0.428501i	0.976779	−	0.214250i	$-0.0687309\pi$
$368$	0	0
$369$	243.816	0.660747
$370$	0	0
$371$	−438.632	−1.18230
$372$	0	0
$373$	563.164i	1.50982i	0.655826	+	0.754912i	$0.272320\pi$
$373$	563.164i	1.50982i	−0.655826	+	0.754912i	$0.727680\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.6816	0.0283333
$378$	0	0
$379$	523.427	1.38107	0.690537	−	0.723297i	$-0.257374\pi$
$379$	523.427	1.38107	0.690537	+	0.723297i	$0.257374\pi$
$380$	0	0
$381$	47.6944i	0.125182i
$382$	0	0
$383$	− 10.0922i	− 0.0263505i	−0.999913	−	0.0131752i	$-0.995806\pi$
$383$	− 10.0922i	− 0.0263505i	0.999913	−	0.0131752i	$-0.00419393\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−807.964	−2.08776
$388$	0	0
$389$	590.772i	1.51869i	0.650686	+	0.759347i	$0.274481\pi$
$389$	590.772i	1.51869i	−0.650686	+	0.759347i	$0.725519\pi$
$390$	0	0
$391$	− 362.497i	− 0.927103i
$392$	0	0
$393$	−141.503	−0.360058
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 143.796i	− 0.362205i	−0.983464	−	0.181103i	$-0.942033\pi$
$397$	− 143.796i	− 0.362205i	0.983464	−	0.181103i	$-0.0579666\pi$
$398$	0	0
$399$	− 1418.29i	− 3.55461i
$400$	0	0
$401$	−189.330	−0.472144	−0.236072	−	0.971736i	$-0.575860\pi$
$401$	−189.330	−0.472144	−0.236072	+	0.971736i	$0.575860\pi$
$402$	0	0
$403$	−94.3646	−0.234155
$404$	0	0
$405$	0	0
$406$	0	0
$407$	427.352i	1.05001i
$408$	0	0
$409$	−575.816	−1.40786	−0.703931	−	0.710268i	$-0.748574\pi$
$409$	−575.816	−1.40786	−0.703931	+	0.710268i	$0.748574\pi$
$410$	0	0
$411$	−1012.39	−2.46323
$412$	0	0
$413$	718.127i	1.73881i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−264.486	−0.634259
$418$	0	0
$419$	539.992	1.28876	0.644381	−	0.764704i	$-0.277115\pi$
$419$	539.992	1.28876	0.644381	+	0.764704i	$0.277115\pi$
$420$	0	0
$421$	− 138.336i	− 0.328590i	−0.986411	−	0.164295i	$-0.947465\pi$
$421$	− 138.336i	− 0.328590i	0.986411	−	0.164295i	$-0.0525349\pi$
$422$	0	0
$423$	− 1748.36i	− 4.13324i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	833.803	1.95270
$428$	0	0
$429$	112.853i	0.263060i
$430$	0	0
$431$	538.128i	1.24856i	0.781202	+	0.624279i	$0.214607\pi$
$431$	538.128i	1.24856i	−0.781202	+	0.624279i	$0.785393\pi$
$432$	0	0
$433$	369.145	0.852529	0.426265	−	0.904599i	$-0.359829\pi$
$433$	369.145	0.852529	0.426265	+	0.904599i	$0.359829\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 472.010i	− 1.08011i
$438$	0	0
$439$	− 211.654i	− 0.482127i	−0.970509	−	0.241063i	$-0.922504\pi$
$439$	− 211.654i	− 0.482127i	0.970509	−	0.241063i	$-0.0774961\pi$
$440$	0	0
$441$	−1298.97	−2.94550
$442$	0	0
$443$	100.817	0.227577	0.113789	−	0.993505i	$-0.463701\pi$
$443$	100.817	0.227577	0.113789	+	0.993505i	$0.463701\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1503.12i	− 3.36269i
$448$	0	0
$449$	−247.145	−0.550435	−0.275217	−	0.961382i	$-0.588750\pi$
$449$	−247.145	−0.550435	−0.275217	+	0.961382i	$0.588750\pi$
$450$	0	0
$451$	−98.2120	−0.217765
$452$	0	0
$453$	− 1191.04i	− 2.62922i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.66479	0.00801924	0.00400962	−	0.999992i	$-0.498724\pi$
$457$	3.66479	0.00801924	0.00400962	+	0.999992i	$0.498724\pi$
$458$	0	0
$459$	−1341.89	−2.92351
$460$	0	0
$461$	− 179.610i	− 0.389609i	−0.980842	−	0.194805i	$-0.937593\pi$
$461$	− 179.610i	− 0.389609i	0.980842	−	0.194805i	$-0.0624073\pi$
$462$	0	0
$463$	630.047i	1.36079i	0.732844	+	0.680397i	$0.238192\pi$
$463$	630.047i	1.36079i	−0.732844	+	0.680397i	$0.761808\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	239.728	0.513335	0.256668	−	0.966500i	$-0.417375\pi$
$467$	239.728	0.513335	0.256668	+	0.966500i	$0.417375\pi$
$468$	0	0
$469$	− 424.253i	− 0.904591i
$470$	0	0
$471$	− 1389.78i	− 2.95070i
$472$	0	0
$473$	325.458	0.688072
$474$	0	0
$475$	0	0
$476$	0	0
$477$	919.374i	1.92741i
$478$	0	0
$479$	448.648i	0.936635i	0.883560	+	0.468317i	$0.155140\pi$
$479$	448.648i	0.936635i	−0.883560	+	0.468317i	$0.844860\pi$
$480$	0	0
$481$	112.301	0.233475
$482$	0	0
$483$	−1113.30	−2.30497
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 105.561i	− 0.216758i	−0.994110	−	0.108379i	$-0.965434\pi$
$487$	− 105.561i	− 0.216758i	0.994110	−	0.108379i	$-0.0345661\pi$
$488$	0	0
$489$	−678.486	−1.38750
$490$	0	0
$491$	813.904	1.65764	0.828822	−	0.559512i	$-0.189011\pi$
$491$	813.904	1.65764	0.828822	+	0.559512i	$0.189011\pi$
$492$	0	0
$493$	− 87.0442i	− 0.176560i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	293.341	0.590223
$498$	0	0
$499$	874.952	1.75341	0.876705	−	0.481028i	$-0.159736\pi$
$499$	874.952	1.75341	0.876705	+	0.481028i	$0.159736\pi$
$500$	0	0
$501$	196.624i	0.392463i
$502$	0	0
$503$	− 684.589i	− 1.36101i	−0.732742	−	0.680506i	$-0.761760\pi$
$503$	− 684.589i	− 1.36101i	0.732742	−	0.680506i	$-0.238240\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−908.749	−1.79240
$508$	0	0
$509$	219.409i	0.431060i	0.976497	+	0.215530i	$0.0691478\pi$
$509$	219.409i	0.431060i	−0.976497	+	0.215530i	$0.930852\pi$
$510$	0	0
$511$	369.804i	0.723687i
$512$	0	0
$513$	−1747.28	−3.40601
$514$	0	0
$515$	0	0
$516$	0	0
$517$	704.261i	1.36221i
$518$	0	0
$519$	521.480i	1.00478i
$520$	0	0
$521$	−875.966	−1.68132	−0.840659	−	0.541565i	$-0.817832\pi$
$521$	−875.966	−1.68132	−0.840659	+	0.541565i	$0.817832\pi$
$522$	0	0
$523$	656.086	1.25447	0.627233	−	0.778831i	$-0.284187\pi$
$523$	656.086	1.25447	0.627233	+	0.778831i	$0.284187\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	768.972i	1.45915i
$528$	0	0
$529$	158.492	0.299606
$530$	0	0
$531$	1505.20	2.83465
$532$	0	0
$533$	25.8086i	0.0484214i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	187.492	0.349146
$538$	0	0
$539$	523.240	0.970760
$540$	0	0
$541$	− 575.556i	− 1.06387i	−0.846784	−	0.531937i	$-0.821464\pi$
$541$	− 575.556i	− 1.06387i	0.846784	−	0.531937i	$-0.178536\pi$
$542$	0	0
$543$	256.648i	0.472648i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	436.027	0.797124	0.398562	−	0.917141i	$-0.369509\pi$
$547$	436.027	0.797124	0.398562	+	0.917141i	$0.369509\pi$
$548$	0	0
$549$	− 1747.66i	− 3.18334i
$550$	0	0
$551$	− 113.341i	− 0.205700i
$552$	0	0
$553$	−308.994	−0.558760
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 87.4939i	− 0.157081i	−0.996911	−	0.0785403i	$-0.974974\pi$
$557$	− 87.4939i	− 0.157081i	0.996911	−	0.0785403i	$-0.0250259\pi$
$558$	0	0
$559$	− 85.5253i	− 0.152997i
$560$	0	0
$561$	919.631	1.63927
$562$	0	0
$563$	625.718	1.11140	0.555700	−	0.831383i	$-0.312450\pi$
$563$	625.718	1.11140	0.555700	+	0.831383i	$0.312450\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2074.51i	3.65875i
$568$	0	0
$569$	963.469	1.69327	0.846634	−	0.532176i	$-0.178626\pi$
$569$	963.469	1.69327	0.846634	+	0.532176i	$0.178626\pi$
$570$	0	0
$571$	−406.665	−0.712197	−0.356099	−	0.934448i	$-0.615893\pi$
$571$	−406.665	−0.712197	−0.356099	+	0.934448i	$0.615893\pi$
$572$	0	0
$573$	1477.85i	2.57915i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	577.307	1.00053	0.500266	−	0.865872i	$-0.333236\pi$
$577$	577.307	1.00053	0.500266	+	0.865872i	$0.333236\pi$
$578$	0	0
$579$	−583.963	−1.00857
$580$	0	0
$581$	992.614i	1.70846i
$582$	0	0
$583$	− 370.335i	− 0.635223i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	287.872	0.490412	0.245206	−	0.969471i	$-0.421144\pi$
$587$	287.872	0.490412	0.245206	+	0.969471i	$0.421144\pi$
$588$	0	0
$589$	1001.28i	1.69997i
$590$	0	0
$591$	− 1934.78i	− 3.27373i
$592$	0	0
$593$	−834.972	−1.40805	−0.704024	−	0.710177i	$-0.748615\pi$
$593$	−834.972	−1.40805	−0.704024	+	0.710177i	$0.748615\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1251.12i	2.09568i
$598$	0	0
$599$	1127.65i	1.88256i	0.337627	+	0.941280i	$0.390376\pi$
$599$	1127.65i	1.88256i	−0.337627	+	0.941280i	$0.609624\pi$
$600$	0	0
$601$	100.810	0.167737	0.0838685	−	0.996477i	$-0.473272\pi$
$601$	100.810	0.167737	0.0838685	+	0.996477i	$0.473272\pi$
$602$	0	0
$603$	−889.237	−1.47469
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 390.092i	− 0.642656i	−0.946968	−	0.321328i	$-0.895871\pi$
$607$	− 390.092i	− 0.642656i	0.946968	−	0.321328i	$-0.104129\pi$
$608$	0	0
$609$	−267.330	−0.438965
$610$	0	0
$611$	185.069	0.302895
$612$	0	0
$613$	283.731i	0.462856i	0.972852	+	0.231428i	$0.0743397\pi$
$613$	283.731i	0.462856i	−0.972852	+	0.231428i	$0.925660\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−59.1339	−0.0958410	−0.0479205	−	0.998851i	$-0.515259\pi$
$617$	−59.1339	−0.0958410	−0.0479205	+	0.998851i	$0.515259\pi$
$618$	0	0
$619$	−451.148	−0.728834	−0.364417	−	0.931236i	$-0.618732\pi$
$619$	−451.148	−0.728834	−0.364417	+	0.931236i	$0.618732\pi$
$620$	0	0
$621$	1371.55i	2.20861i
$622$	0	0
$623$	− 204.832i	− 0.328784i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1197.46	1.90982
$628$	0	0
$629$	− 915.139i	− 1.45491i
$630$	0	0
$631$	181.480i	0.287608i	0.989606	+	0.143804i	$0.0459334\pi$
$631$	181.480i	0.287608i	−0.989606	+	0.143804i	$0.954067\pi$
$632$	0	0
$633$	−1960.09	−3.09652
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 137.499i	− 0.215855i
$638$	0	0
$639$	− 614.844i	− 0.962197i
$640$	0	0
$641$	−695.168	−1.08450	−0.542252	−	0.840216i	$-0.682428\pi$
$641$	−695.168	−1.08450	−0.542252	+	0.840216i	$0.682428\pi$
$642$	0	0
$643$	−448.806	−0.697988	−0.348994	−	0.937125i	$-0.613477\pi$
$643$	−448.806	−0.697988	−0.348994	+	0.937125i	$0.613477\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1095.09i	1.69256i	0.532738	+	0.846280i	$0.321163\pi$
$647$	1095.09i	1.69256i	−0.532738	+	0.846280i	$0.678837\pi$
$648$	0	0
$649$	−606.313	−0.934226
$650$	0	0
$651$	2361.66	3.62775
$652$	0	0
$653$	965.844i	1.47909i	0.673109	+	0.739543i	$0.264958\pi$
$653$	965.844i	1.47909i	−0.673109	+	0.739543i	$0.735042\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	775.111	1.17977
$658$	0	0
$659$	−261.395	−0.396654	−0.198327	−	0.980136i	$-0.563551\pi$
$659$	−261.395	−0.396654	−0.198327	+	0.980136i	$0.563551\pi$
$660$	0	0
$661$	1095.32i	1.65707i	0.559937	+	0.828535i	$0.310825\pi$
$661$	1095.32i	1.65707i	−0.559937	+	0.828535i	$0.689175\pi$
$662$	0	0
$663$	− 241.665i	− 0.364502i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−88.9679	−0.133385
$668$	0	0
$669$	− 501.154i	− 0.749109i
$670$	0	0
$671$	703.978i	1.04915i
$672$	0	0
$673$	1033.10	1.53507	0.767534	−	0.641009i	$-0.221484\pi$
$673$	1033.10	1.53507	0.767534	+	0.641009i	$0.221484\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1014.89i	1.49909i	0.661951	+	0.749547i	$0.269729\pi$
$677$	1014.89i	1.49909i	−0.661951	+	0.749547i	$0.730271\pi$
$678$	0	0
$679$	522.497i	0.769510i
$680$	0	0
$681$	−251.827	−0.369790
$682$	0	0
$683$	−125.851	−0.184262	−0.0921308	−	0.995747i	$-0.529368\pi$
$683$	−125.851	−0.184262	−0.0921308	+	0.995747i	$0.529368\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1372.30i	− 1.99753i
$688$	0	0
$689$	−97.3184	−0.141246
$690$	0	0
$691$	547.499	0.792329	0.396164	−	0.918180i	$-0.370341\pi$
$691$	547.499	0.792329	0.396164	+	0.918180i	$0.370341\pi$
$692$	0	0
$693$	− 1999.93i	− 2.88590i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	210.313	0.301740
$698$	0	0
$699$	741.238	1.06043
$700$	0	0
$701$	− 749.070i	− 1.06857i	−0.845303	−	0.534287i	$-0.820580\pi$
$701$	− 749.070i	− 1.06857i	0.845303	−	0.534287i	$-0.179420\pi$
$702$	0	0
$703$	− 1191.61i	− 1.69503i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−423.928	−0.599616
$708$	0	0
$709$	356.147i	0.502323i	0.967945	+	0.251161i	$0.0808125\pi$
$709$	356.147i	0.502323i	−0.967945	+	0.251161i	$0.919187\pi$
$710$	0	0
$711$	647.654i	0.910905i
$712$	0	0
$713$	785.966	1.10234
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.1441i	0.0671466i
$718$	0	0
$719$	− 487.978i	− 0.678689i	−0.940662	−	0.339345i	$-0.889795\pi$
$719$	− 487.978i	− 0.678689i	0.940662	−	0.339345i	$-0.110205\pi$
$720$	0	0
$721$	−98.1395	−0.136116
$722$	0	0
$723$	1446.40	2.00055
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1258.53i	− 1.73113i	−0.500795	−	0.865566i	$-0.666959\pi$
$727$	− 1258.53i	− 1.73113i	0.500795	−	0.865566i	$-0.333041\pi$
$728$	0	0
$729$	787.301	1.07997
$730$	0	0
$731$	−696.941	−0.953408
$732$	0	0
$733$	− 534.408i	− 0.729069i	−0.931190	−	0.364535i	$-0.881228\pi$
$733$	− 534.408i	− 0.729069i	0.931190	−	0.364535i	$-0.118772\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	358.196	0.486019
$738$	0	0
$739$	−988.704	−1.33789	−0.668947	−	0.743310i	$-0.733255\pi$
$739$	−988.704	−1.33789	−0.668947	+	0.743310i	$0.733255\pi$
$740$	0	0
$741$	− 314.673i	− 0.424660i
$742$	0	0
$743$	227.550i	0.306259i	0.988206	+	0.153129i	$0.0489351\pi$
$743$	227.550i	0.306259i	−0.988206	+	0.153129i	$0.951065\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2080.52	2.78517
$748$	0	0
$749$	1098.92i	1.46718i
$750$	0	0
$751$	− 642.475i	− 0.855492i	−0.903899	−	0.427746i	$-0.859308\pi$
$751$	− 642.475i	− 0.855492i	0.903899	−	0.427746i	$-0.140692\pi$
$752$	0	0
$753$	2289.44	3.04042
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1087.75i	− 1.43693i	−0.695565	−	0.718463i	$-0.744846\pi$
$757$	− 1087.75i	− 1.43693i	0.695565	−	0.718463i	$-0.255154\pi$
$758$	0	0
$759$	− 939.955i	− 1.23841i
$760$	0	0
$761$	−633.955	−0.833055	−0.416528	−	0.909123i	$-0.636753\pi$
$761$	−633.955	−0.833055	−0.416528	+	0.909123i	$0.636753\pi$
$762$	0	0
$763$	1383.43	1.81314
$764$	0	0
$765$	0	0
$766$	0	0
$767$	159.330i	0.207731i
$768$	0	0
$769$	−1038.32	−1.35023	−0.675113	−	0.737714i	$-0.735905\pi$
$769$	−1038.32	−1.35023	−0.675113	+	0.737714i	$0.735905\pi$
$770$	0	0
$771$	−1736.10	−2.25175
$772$	0	0
$773$	− 293.874i	− 0.380174i	−0.981767	−	0.190087i	$-0.939123\pi$
$773$	− 293.874i	− 0.380174i	0.981767	−	0.190087i	$-0.0608769\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2810.57	−3.61721
$778$	0	0
$779$	273.850	0.351540
$780$	0	0
$781$	247.667i	0.317115i
$782$	0	0
$783$	329.341i	0.420614i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−666.479	−0.846860	−0.423430	−	0.905929i	$-0.639174\pi$
$787$	−666.479	−0.846860	−0.423430	+	0.905929i	$0.639174\pi$
$788$	0	0
$789$	− 343.755i	− 0.435684i
$790$	0	0
$791$	1170.11i	1.47927i
$792$	0	0
$793$	184.994	0.233284
$794$	0	0
$795$	0	0
$796$	0	0
$797$	998.973i	1.25342i	0.779254	+	0.626708i	$0.215598\pi$
$797$	998.973i	1.25342i	−0.779254	+	0.626708i	$0.784402\pi$
$798$	0	0
$799$	− 1508.12i	− 1.88751i
$800$	0	0
$801$	−429.330	−0.535992
$802$	0	0
$803$	−312.225	−0.388823
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 747.469i	− 0.926232i
$808$	0	0
$809$	813.330	1.00535	0.502676	−	0.864475i	$-0.332349\pi$
$809$	813.330	1.00535	0.502676	+	0.864475i	$0.332349\pi$
$810$	0	0
$811$	−264.156	−0.325716	−0.162858	−	0.986650i	$-0.552071\pi$
$811$	−264.156	−0.325716	−0.162858	+	0.986650i	$0.552071\pi$
$812$	0	0
$813$	1472.33i	1.81099i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−907.492	−1.11076
$818$	0	0
$819$	−525.551	−0.641698
$820$	0	0
$821$	− 145.007i	− 0.176622i	−0.996093	−	0.0883112i	$-0.971853\pi$
$821$	− 145.007i	− 0.176622i	0.996093	−	0.0883112i	$-0.0281470\pi$
$822$	0	0
$823$	1098.39i	1.33462i	0.744779	+	0.667311i	$0.232555\pi$
$823$	1098.39i	1.33462i	−0.744779	+	0.667311i	$0.767445\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1329.67	−1.60782	−0.803910	−	0.594751i	$-0.797251\pi$
$827$	−1329.67	−1.60782	−0.803910	+	0.594751i	$0.797251\pi$
$828$	0	0
$829$	734.180i	0.885621i	0.896615	+	0.442811i	$0.146019\pi$
$829$	734.180i	0.885621i	−0.896615	+	0.442811i	$0.853981\pi$
$830$	0	0
$831$	− 968.463i	− 1.16542i
$832$	0	0
$833$	−1120.47	−1.34511
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2909.49i	− 3.47609i
$838$	0	0
$839$	− 42.3127i	− 0.0504323i	−0.999682	−	0.0252162i	$-0.991973\pi$
$839$	− 42.3127i	− 0.0504323i	0.999682	−	0.0252162i	$-0.00802741\pi$
$840$	0	0
$841$	819.637	0.974598
$842$	0	0
$843$	−2647.58	−3.14066
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 454.763i	− 0.536910i
$848$	0	0
$849$	313.492	0.369248
$850$	0	0
$851$	−935.364	−1.09913
$852$	0	0
$853$	− 604.700i	− 0.708910i	−0.935073	−	0.354455i	$-0.884666\pi$
$853$	− 604.700i	− 0.708910i	0.935073	−	0.354455i	$-0.115334\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1578.63	1.84204	0.921018	−	0.389519i	$-0.127359\pi$
$857$	1578.63	1.84204	0.921018	+	0.389519i	$0.127359\pi$
$858$	0	0
$859$	−558.355	−0.650006	−0.325003	−	0.945713i	$-0.605365\pi$
$859$	−558.355	−0.650006	−0.325003	+	0.945713i	$0.605365\pi$
$860$	0	0
$861$	− 645.911i	− 0.750187i
$862$	0	0
$863$	− 311.550i	− 0.361008i	−0.983574	−	0.180504i	$-0.942227\pi$
$863$	− 311.550i	− 0.361008i	0.983574	−	0.180504i	$-0.0577729\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−364.585	−0.420514
$868$	0	0
$869$	− 260.883i	− 0.300211i
$870$	0	0
$871$	− 94.1283i	− 0.108069i
$872$	0	0
$873$	1095.16	1.25447
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1054.50i	− 1.20239i	−0.799101	−	0.601197i	$-0.794691\pi$
$877$	− 1054.50i	− 1.20239i	0.799101	−	0.601197i	$-0.205309\pi$
$878$	0	0
$879$	− 69.5028i	− 0.0790703i
$880$	0	0
$881$	1053.16	1.19541	0.597705	−	0.801716i	$-0.296079\pi$
$881$	1053.16	1.19541	0.597705	+	0.801716i	$0.296079\pi$
$882$	0	0
$883$	229.522	0.259934	0.129967	−	0.991518i	$-0.458513\pi$
$883$	229.522	0.259934	0.129967	+	0.991518i	$0.458513\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1422.07i	1.60324i	0.597837	+	0.801618i	$0.296027\pi$
$887$	1422.07i	1.60324i	−0.597837	+	0.801618i	$0.703973\pi$
$888$	0	0
$889$	89.4691	0.100640
$890$	0	0
$891$	−1751.50	−1.96577
$892$	0	0
$893$	− 1963.73i	− 2.19903i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−247.006	−0.275369
$898$	0	0
$899$	188.729	0.209932
$900$	0	0
$901$	793.042i	0.880180i
$902$	0	0
$903$	2140.44i	2.37037i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−222.388	−0.245191	−0.122596	−	0.992457i	$-0.539122\pi$
$907$	−222.388	−0.245191	−0.122596	+	0.992457i	$0.539122\pi$
$908$	0	0
$909$	888.556i	0.977509i
$910$	0	0
$911$	724.855i	0.795669i	0.917457	+	0.397835i	$0.130238\pi$
$911$	724.855i	0.795669i	−0.917457	+	0.397835i	$0.869762\pi$
$912$	0	0
$913$	−838.061	−0.917920
$914$	0	0
$915$	0	0
$916$	0	0
$917$	265.443i	0.289469i
$918$	0	0
$919$	260.972i	0.283974i	0.989869	+	0.141987i	$0.0453491\pi$
$919$	260.972i	0.283974i	−0.989869	+	0.141987i	$0.954651\pi$
$920$	0	0
$921$	2558.74	2.77822
$922$	0	0
$923$	65.0830	0.0705124
$924$	0	0
$925$	0	0
$926$	0	0
$927$	205.701i	0.221900i
$928$	0	0
$929$	1004.18	1.08093	0.540465	−	0.841366i	$-0.318248\pi$
$929$	1004.18	1.08093	0.540465	+	0.841366i	$0.318248\pi$
$930$	0	0
$931$	−1458.98	−1.56711
$932$	0	0
$933$	622.801i	0.667525i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−335.525	−0.358085	−0.179042	−	0.983841i	$-0.557300\pi$
$937$	−335.525	−0.358085	−0.179042	+	0.983841i	$0.557300\pi$
$938$	0	0
$939$	2176.59	2.31799
$940$	0	0
$941$	− 1741.43i	− 1.85062i	−0.379210	−	0.925311i	$-0.623804\pi$
$941$	− 1741.43i	− 1.85062i	0.379210	−	0.925311i	$-0.376196\pi$
$942$	0	0
$943$	− 214.961i	− 0.227954i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1597.22	1.68661	0.843306	−	0.537434i	$-0.180606\pi$
$947$	1597.22	1.68661	0.843306	+	0.537434i	$0.180606\pi$
$948$	0	0
$949$	82.0478i	0.0864571i
$950$	0	0
$951$	2109.44i	2.21812i
$952$	0	0
$953$	−651.642	−0.683780	−0.341890	−	0.939740i	$-0.611067\pi$
$953$	−651.642	−0.683780	−0.341890	+	0.939740i	$0.611067\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 225.706i	− 0.235847i
$958$	0	0
$959$	1899.12i	1.98032i
$960$	0	0
$961$	−706.285	−0.734948
$962$	0	0
$963$	2303.34	2.39184
$964$	0	0
$965$	0	0
$966$	0	0
$967$	221.885i	0.229457i	0.993397	+	0.114729i	$0.0365999\pi$
$967$	221.885i	0.229457i	−0.993397	+	0.114729i	$0.963400\pi$
$968$	0	0
$969$	−2564.26	−2.64629
$970$	0	0
$971$	1894.66	1.95125	0.975624	−	0.219450i	$-0.0704262\pi$
$971$	1894.66	1.95125	0.975624	+	0.219450i	$0.0704262\pi$
$972$	0	0
$973$	496.145i	0.509912i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	818.810	0.838086	0.419043	−	0.907966i	$-0.362366\pi$
$977$	818.810	0.838086	0.419043	+	0.907966i	$0.362366\pi$
$978$	0	0
$979$	172.939	0.176649
$980$	0	0
$981$	− 2899.67i	− 2.95583i
$982$	0	0
$983$	− 1141.70i	− 1.16145i	−0.814101	−	0.580723i	$-0.802770\pi$
$983$	− 1141.70i	− 1.16145i	0.814101	−	0.580723i	$-0.197230\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4631.72	−4.69273
$988$	0	0
$989$	712.343i	0.720266i
$990$	0	0
$991$	− 336.162i	− 0.339215i	−0.985512	−	0.169607i	$-0.945750\pi$
$991$	− 336.162i	− 0.339215i	0.985512	−	0.169607i	$-0.0542500\pi$
$992$	0	0
$993$	1492.44	1.50296
$994$	0	0
$995$	0	0
$996$	0	0
$997$	934.014i	0.936825i	0.883510	+	0.468412i	$0.155174\pi$
$997$	934.014i	0.936825i	−0.883510	+	0.468412i	$0.844826\pi$
$998$	0	0
$999$	3462.52i	3.46599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.g.j.351.2		8
4.3	odd	2	inner	1600.3.g.j.351.7		8
5.2	odd	4		1600.3.e.i.799.7		8
5.3	odd	4		1600.3.e.j.799.1		8
5.4	even	2		320.3.g.b.31.7	yes	8
8.3	odd	2	inner	1600.3.g.j.351.1		8
8.5	even	2	inner	1600.3.g.j.351.8		8
15.14	odd	2		2880.3.g.e.991.5		8
20.3	even	4		1600.3.e.i.799.8		8
20.7	even	4		1600.3.e.j.799.2		8
20.19	odd	2		320.3.g.b.31.1	✓	8
40.3	even	4		1600.3.e.i.799.1		8
40.13	odd	4		1600.3.e.j.799.8		8
40.19	odd	2		320.3.g.b.31.8	yes	8
40.27	even	4		1600.3.e.j.799.7		8
40.29	even	2		320.3.g.b.31.2	yes	8
40.37	odd	4		1600.3.e.i.799.2		8
60.59	even	2		2880.3.g.e.991.8		8
80.19	odd	4		1280.3.b.e.511.1		8
80.29	even	4		1280.3.b.e.511.7		8
80.59	odd	4		1280.3.b.e.511.8		8
80.69	even	4		1280.3.b.e.511.2		8
120.29	odd	2		2880.3.g.e.991.1		8
120.59	even	2		2880.3.g.e.991.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.g.b.31.1	✓	8	20.19	odd	2
320.3.g.b.31.2	yes	8	40.29	even	2
320.3.g.b.31.7	yes	8	5.4	even	2
320.3.g.b.31.8	yes	8	40.19	odd	2
1280.3.b.e.511.1		8	80.19	odd	4
1280.3.b.e.511.2		8	80.69	even	4
1280.3.b.e.511.7		8	80.29	even	4
1280.3.b.e.511.8		8	80.59	odd	4
1600.3.e.i.799.1		8	40.3	even	4
1600.3.e.i.799.2		8	40.37	odd	4
1600.3.e.i.799.7		8	5.2	odd	4
1600.3.e.i.799.8		8	20.3	even	4
1600.3.e.j.799.1		8	5.3	odd	4
1600.3.e.j.799.2		8	20.7	even	4
1600.3.e.j.799.7		8	40.27	even	4
1600.3.e.j.799.8		8	40.13	odd	4
1600.3.g.j.351.1		8	8.3	odd	2	inner
1600.3.g.j.351.2		8	1.1	even	1	trivial
1600.3.g.j.351.7		8	4.3	odd	2	inner
1600.3.g.j.351.8		8	8.5	even	2	inner
2880.3.g.e.991.1		8	120.29	odd	2
2880.3.g.e.991.4		8	120.59	even	2
2880.3.g.e.991.5		8	15.14	odd	2
2880.3.g.e.991.8		8	60.59	even	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.55269 q^{3} +10.4162i q^{7} +21.8324 q^{9} +O(q^{10})\)	\(q-5.55269 q^{3} +10.4162i q^{7} +21.8324 q^{9} -8.79436 q^{11} +2.31102i q^{13} +18.8324 q^{17} +24.5218 q^{19} -57.8380i q^{21} -19.2486i q^{23} -71.2544 q^{27} -4.62204i q^{29} +40.8324i q^{31} +48.8324 q^{33} -48.5939i q^{37} -12.8324i q^{39} +11.1676 q^{41} -37.0076 q^{43} -80.0810i q^{47} -59.4972 q^{49} -104.571 q^{51} +42.1105i q^{53} -136.162 q^{57} +68.9433 q^{59} -80.0487i q^{61} +227.411i q^{63} -40.7301 q^{67} +106.882i q^{69} -28.1620i q^{71} +35.5028 q^{73} -91.6038i q^{77} +29.6648i q^{79} +199.162 q^{81} +95.2952 q^{83} +25.6648i q^{87} -19.6648 q^{89} -24.0721 q^{91} -226.730i q^{93} +50.1620 q^{97} -192.002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 56 q^{9}+O(q^{10}) \) 8 * q + 56 * q^9	\( 8 q + 56 q^{9} + 32 q^{17} + 272 q^{33} + 208 q^{41} - 120 q^{49} - 496 q^{57} + 640 q^{73} + 1000 q^{81} + 80 q^{89} - 192 q^{97}+O(q^{100}) \) 8 * q + 56 * q^9 + 32 * q^17 + 272 * q^33 + 208 * q^41 - 120 * q^49 - 496 * q^57 + 640 * q^73 + 1000 * q^81 + 80 * q^89 - 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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