LMFDB - Embedded newform 2850.2.d.n.799.1

Newspace parameters

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.d (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$22.7573645761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.n.799.2

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} +2.00000i q^{28} -1.00000i q^{32} -2.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} +2.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +8.00000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +2.00000i q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +8.00000i q^{82} +16.0000i q^{83} -2.00000 q^{84} +6.00000 q^{86} -2.00000i q^{88} -16.0000 q^{89} -8.00000i q^{92} -8.00000 q^{94} +1.00000 q^{96} +8.00000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) $	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} - 4 q^{34} + 2 q^{36} - 16 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{49} + 4 q^{51} - 2 q^{54} + 4 q^{56} + 16 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} + 16 q^{71} + 8 q^{74} + 2 q^{76} + 16 q^{79} + 2 q^{81} - 4 q^{84} + 12 q^{86} - 32 q^{89} - 16 q^{94} + 2 q^{96} + 4 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 1.00000i	− 0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	2.00000i	0.426401i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	2.00000i	0.377964i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 2.00000i	− 0.348155i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	− 2.00000i	− 0.308607i
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	2.00000i	0.242536i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	1.00000	0.114708
$77$	4.00000i	0.455842i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.00000i	0.883452i
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	6.00000	0.646997
$87$	0	0
$88$	− 2.00000i	− 0.213201i
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	0	0
$92$	− 8.00000i	− 0.834058i
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	− 3.00000i	− 0.303046i
$99$	2.00000	0.201008
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	0	0
$105$	0	0
$106$	10.0000	0.971286
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	1.00000i	0.0962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 2.00000i	− 0.188982i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	0	0
$117$	0	0
$118$	− 8.00000i	− 0.736460i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 2.00000i	− 0.181071i
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	0	0
$126$	2.00000	0.178174
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	2.00000i	0.174078i
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	2.00000	0.171499
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	8.00000i	0.681005i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	− 8.00000i	− 0.671345i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	3.00000i	0.247436i
$148$	− 4.00000i	− 0.328798i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	2.00000i	0.161690i
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	16.0000	1.26098
$162$	− 1.00000i	− 0.0785674i
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	16.0000	1.24184
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	2.00000i	0.154303i
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 6.00000i	− 0.457496i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	8.00000i	0.601317i
$178$	16.0000i	1.19925i
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	−8.00000	−0.589768
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	8.00000i	0.583460i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	− 2.00000i	− 0.142134i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−12.0000	−0.836080
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 10.0000i	− 0.686803i
$213$	8.00000i	0.548151i
$214$	20.0000	1.36717
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.00000i	0.135457i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	4.00000i	0.268462i
$223$	− 12.0000i	− 0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$	− 12.0000i	− 0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	2.00000	0.133038
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	1.00000i	0.0662266i
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	−8.00000	−0.520756
$237$	8.00000i	0.519656i
$238$	4.00000i	0.259281i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000i	0.449977i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−8.00000	−0.510061
$247$	0	0
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	− 2.00000i	− 0.125988i
$253$	− 16.0000i	− 1.00591i
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	6.00000i	0.373544i
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	6.00000i	0.370681i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	2.00000	0.122628
$267$	− 16.0000i	− 0.979184i
$268$	0	0
$269$	−8.00000	−0.487769	−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000	−0.487769	−0.243884	+	0.969804i	$0.578422\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	8.00000	0.481543
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	8.00000i	0.479808i
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	− 8.00000i	− 0.476393i
$283$	2.00000i	0.118888i	0.998232	+	0.0594438i	$0.0189327\pi$
$283$	2.00000i	0.118888i	−0.998232	+	0.0594438i	$0.981067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	0	0
$287$	16.0000i	0.944450i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−8.00000	−0.468968
$292$	− 2.00000i	− 0.117041i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−4.00000	−0.232495
$297$	2.00000i	0.116052i
$298$	2.00000i	0.115857i
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	8.00000i	0.460348i
$303$	2.00000i	0.114897i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	− 24.0000i	− 1.36975i	−0.728659	−	0.684876i	$-0.759856\pi$
$307$	− 24.0000i	− 1.36975i	0.728659	−	0.684876i	$-0.240144\pi$
$308$	− 4.00000i	− 0.227921i
$309$	12.0000	0.682656
$310$	0	0
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	− 18.0000i	− 1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$	− 18.0000i	− 1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−8.00000	−0.450035
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	10.0000i	0.560772i
$319$	0	0
$320$	0	0
$321$	−20.0000	−1.11629
$322$	− 16.0000i	− 0.891645i
$323$	2.00000i	0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	10.0000	0.553849
$327$	− 2.00000i	− 0.110600i
$328$	− 8.00000i	− 0.441726i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 16.0000i	− 0.878114i
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	− 16.0000i	− 0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$	− 16.0000i	− 0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$	− 13.0000i	− 0.707107i
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 0.0540738i
$343$	− 20.0000i	− 1.07990i
$344$	−6.00000	−0.323498
$345$	0	0
$346$	6.00000	0.322562
$347$	− 8.00000i	− 0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$	− 8.00000i	− 0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	0	0
$352$	2.00000i	0.106600i
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	16.0000	0.847998
$357$	− 4.00000i	− 0.211702i
$358$	− 8.00000i	− 0.422813i
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.00000i	0.315353i
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	8.00000i	0.417029i
$369$	8.00000	0.416463
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	2.00000i	0.102869i
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 10.0000i	− 0.511645i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	4.00000	0.203595
$387$	− 6.00000i	− 0.304997i
$388$	− 8.00000i	− 0.406138i
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	3.00000i	0.151523i
$393$	− 6.00000i	− 0.302660i
$394$	10.0000	0.503793
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	− 4.00000i	− 0.200502i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	2.00000i	0.0990148i
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	12.0000i	0.591198i
$413$	− 16.0000i	− 0.787309i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	0	0
$417$	− 8.00000i	− 0.391762i
$418$	− 2.00000i	− 0.0978232i
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	− 4.00000i	− 0.194717i
$423$	8.00000i	0.388973i
$424$	−10.0000	−0.485643
$425$	0	0
$426$	8.00000	0.387601
$427$	− 4.00000i	− 0.193574i
$428$	− 20.0000i	− 0.966736i
$429$	0	0
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	12.0000i	0.576683i	0.957528	+	0.288342i	$0.0931039\pi$
$433$	12.0000i	0.576683i	−0.957528	+	0.288342i	$0.906896\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 8.00000i	− 0.382692i
$438$	2.00000i	0.0955637i
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−12.0000	−0.568216
$447$	− 2.00000i	− 0.0945968i
$448$	2.00000i	0.0944911i
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	− 2.00000i	− 0.0940721i
$453$	− 8.00000i	− 0.375873i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	1.00000	0.0468293
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	10.0000i	0.467269i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	4.00000i	0.186097i
$463$	− 18.0000i	− 0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$	− 18.0000i	− 0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.00000	0.276465
$472$	8.00000i	0.368230i
$473$	− 12.0000i	− 0.551761i
$474$	8.00000	0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	− 10.0000i	− 0.457869i
$478$	6.00000i	0.274434i
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	0	0
$482$	− 10.0000i	− 0.455488i
$483$	16.0000i	0.728025i
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	2.00000i	0.0905357i
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	8.00000i	0.360668i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.0000i	− 0.717698i
$498$	16.0000i	0.716977i
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	− 6.00000i	− 0.267793i
$503$	− 28.0000i	− 1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$	− 28.0000i	− 1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−16.0000	−0.711287
$507$	13.0000i	0.577350i
$508$	− 16.0000i	− 0.709885i
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	30.0000	1.32324
$515$	0	0
$516$	6.00000	0.264135
$517$	16.0000i	0.703679i
$518$	− 8.00000i	− 0.351500i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	0	0
$523$	− 28.0000i	− 1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$	− 28.0000i	− 1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	12.0000	0.523225
$527$	0	0
$528$	− 2.00000i	− 0.0870388i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−8.00000	−0.347170
$532$	− 2.00000i	− 0.0867110i
$533$	0	0
$534$	−16.0000	−0.692388
$535$	0	0
$536$	0	0
$537$	8.00000i	0.345225i
$538$	8.00000i	0.344904i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	4.00000i	0.171815i
$543$	− 6.00000i	− 0.257485i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	− 40.0000i	− 1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$	− 40.0000i	− 1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	− 8.00000i	− 0.340503i
$553$	− 16.0000i	− 0.680389i
$554$	−14.0000	−0.594803
$555$	0	0
$556$	8.00000	0.339276
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−4.00000	−0.168880
$562$	20.0000i	0.843649i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	2.00000	0.0840663
$567$	− 2.00000i	− 0.0839921i
$568$	8.00000i	0.335673i
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	10.0000i	0.417756i
$574$	16.0000	0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−4.00000	−0.166234
$580$	0	0
$581$	32.0000	1.32758
$582$	8.00000i	0.331611i
$583$	− 20.0000i	− 0.828315i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	22.0000	0.908812
$587$	− 44.0000i	− 1.81607i	−0.418890	−	0.908037i	$-0.637581\pi$
$587$	− 44.0000i	− 1.81607i	0.418890	−	0.908037i	$-0.362419\pi$
$588$	− 3.00000i	− 0.123718i
$589$	0	0
$590$	0	0
$591$	−10.0000	−0.411345
$592$	4.00000i	0.164399i
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	2.00000	0.0819232
$597$	4.00000i	0.163709i
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	− 12.0000i	− 0.489083i
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	2.00000	0.0812444
$607$	44.0000i	1.78590i	0.450151	+	0.892952i	$0.351370\pi$
$607$	44.0000i	1.78590i	−0.450151	+	0.892952i	$0.648630\pi$
$608$	1.00000i	0.0405554i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	46.0000i	1.85792i	0.370177	+	0.928961i	$0.379297\pi$
$613$	46.0000i	1.85792i	−0.370177	+	0.928961i	$0.620703\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	−4.00000	−0.161165
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	− 12.0000i	− 0.482711i
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	14.0000i	0.561349i
$623$	32.0000i	1.28205i
$624$	0	0
$625$	0	0
$626$	−18.0000	−0.719425
$627$	2.00000i	0.0798723i
$628$	6.00000i	0.239426i
$629$	8.00000	0.318981
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	8.00000i	0.318223i
$633$	4.00000i	0.158986i
$634$	6.00000	0.238290
$635$	0	0
$636$	10.0000	0.396526
$637$	0	0
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	20.0000i	0.789337i
$643$	− 46.0000i	− 1.81406i	−0.421063	−	0.907031i	$-0.638343\pi$
$643$	− 46.0000i	− 1.81406i	0.421063	−	0.907031i	$-0.361657\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	2.00000	0.0786889
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	1.00000i	0.0392837i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	− 10.0000i	− 0.391630i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−8.00000	−0.312348
$657$	− 2.00000i	− 0.0780274i
$658$	16.0000i	0.623745i
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	12.0000i	0.466393i
$663$	0	0
$664$	−16.0000	−0.620920
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	−4.00000	−0.154418
$672$	− 2.00000i	− 0.0771517i
$673$	16.0000i	0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$	16.0000i	0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−13.0000	−0.500000
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	2.00000i	0.0768095i
$679$	16.0000	0.614024
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 10.0000i	− 0.381524i
$688$	6.00000i	0.228748i
$689$	0	0
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	− 6.00000i	− 0.228086i
$693$	− 4.00000i	− 0.151947i
$694$	−8.00000	−0.303676
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	− 26.0000i	− 0.984115i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	2.00000	0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	− 4.00000i	− 0.150435i
$708$	− 8.00000i	− 0.300658i
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 16.0000i	− 0.599625i
$713$	0	0
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−8.00000	−0.298974
$717$	− 6.00000i	− 0.224074i
$718$	− 30.0000i	− 1.11959i
$719$	−22.0000	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$719$	−22.0000	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	− 1.00000i	− 0.0372161i
$723$	10.0000i	0.371904i
$724$	6.00000	0.222988
$725$	0	0
$726$	−7.00000	−0.259794
$727$	− 22.0000i	− 0.815935i	−0.912996	−	0.407967i	$-0.866238\pi$
$727$	− 22.0000i	− 0.815935i	0.912996	−	0.407967i	$-0.133762\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	− 2.00000i	− 0.0739221i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	8.00000	0.294884
$737$	0	0
$738$	− 8.00000i	− 0.294484i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	− 20.0000i	− 0.734223i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	−32.0000	−1.17160
$747$	− 16.0000i	− 0.585409i
$748$	− 4.00000i	− 0.146254i
$749$	40.0000	1.46157
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	− 8.00000i	− 0.291730i
$753$	6.00000i	0.218652i
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	− 28.0000i	− 1.01701i
$759$	16.0000	0.580763
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	16.0000i	0.579619i
$763$	4.00000i	0.144810i
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	− 4.00000i	− 0.143963i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	−8.00000	−0.287183
$777$	8.00000i	0.286998i
$778$	14.0000i	0.501924i
$779$	8.00000	0.286630
$780$	0	0
$781$	−16.0000	−0.572525
$782$	− 16.0000i	− 0.572159i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	−6.00000	−0.214013
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	− 10.0000i	− 0.356235i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	4.00000	0.142224
$792$	2.00000i	0.0710669i
$793$	0	0
$794$	−6.00000	−0.212932
$795$	0	0
$796$	−4.00000	−0.141776
$797$	− 22.0000i	− 0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	− 22.0000i	− 0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$	2.00000i	0.0707992i
$799$	−16.0000	−0.566039
$800$	0	0
$801$	16.0000	0.565332
$802$	12.0000i	0.423735i
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8.00000i	− 0.281613i
$808$	2.00000i	0.0703598i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	− 4.00000i	− 0.140286i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 6.00000i	− 0.209913i
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	18.0000i	0.627822i
$823$	− 34.0000i	− 1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$	− 34.0000i	− 1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	−16.0000	−0.556711
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	8.00000i	0.278019i
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	30.0000i	1.03633i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	26.0000i	0.896019i
$843$	− 20.0000i	− 0.688837i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	14.0000i	0.481046i
$848$	10.0000i	0.343401i
$849$	−2.00000	−0.0686398
$850$	0	0
$851$	−32.0000	−1.09695
$852$	− 8.00000i	− 0.274075i
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−20.0000	−0.683586
$857$	− 30.0000i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$	− 30.0000i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	− 20.0000i	− 0.681203i
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	12.0000	0.407777
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	0	0
$872$	− 2.00000i	− 0.0677285i
$873$	− 8.00000i	− 0.270759i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	2.00000	0.0675737
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	16.0000i	0.539974i
$879$	−22.0000	−0.742042
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	3.00000i	0.101015i
$883$	18.0000i	0.605748i	0.953031	+	0.302874i	$0.0979462\pi$
$883$	18.0000i	0.605748i	−0.953031	+	0.302874i	$0.902054\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	− 4.00000i	− 0.134231i
$889$	32.0000	1.07325
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	12.0000i	0.401790i
$893$	8.00000i	0.267710i
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	24.0000i	0.800890i
$899$	0	0
$900$	0	0
$901$	20.0000	0.666297
$902$	− 16.0000i	− 0.532742i
$903$	12.0000i	0.399335i
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	4.00000i	0.132745i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	− 32.0000i	− 1.05905i
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	12.0000i	0.396275i
$918$	2.00000i	0.0660098i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	6.00000i	0.197599i
$923$	0	0
$924$	4.00000	0.131590
$925$	0	0
$926$	−18.0000	−0.591517
$927$	12.0000i	0.394132i
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	− 6.00000i	− 0.196537i
$933$	− 14.0000i	− 0.458339i
$934$	20.0000	0.654420
$935$	0	0
$936$	0	0
$937$	− 58.0000i	− 1.89478i	−0.320085	−	0.947389i	$-0.603712\pi$
$937$	− 58.0000i	− 1.89478i	0.320085	−	0.947389i	$-0.396288\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	16.0000	0.521585	0.260793	−	0.965395i	$-0.416016\pi$
$941$	16.0000	0.521585	0.260793	+	0.965395i	$0.416016\pi$
$942$	− 6.00000i	− 0.195491i
$943$	− 64.0000i	− 2.08413i
$944$	8.00000	0.260378
$945$	0	0
$946$	−12.0000	−0.390154
$947$	− 32.0000i	− 1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$	− 32.0000i	− 1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	− 8.00000i	− 0.259828i
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	− 4.00000i	− 0.129641i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	− 30.0000i	− 0.969256i
$959$	36.0000	1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 20.0000i	− 0.644491i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	16.0000	0.514792
$967$	− 6.00000i	− 0.192947i	−0.995336	−	0.0964735i	$-0.969244\pi$
$967$	− 6.00000i	− 0.192947i	0.995336	−	0.0964735i	$-0.0307563\pi$
$968$	− 7.00000i	− 0.224989i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	16.0000i	0.512936i
$974$	20.0000	0.640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	10.0000i	0.319765i
$979$	32.0000	1.02272
$980$	0	0
$981$	2.00000	0.0638551
$982$	30.0000i	0.957338i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	−16.0000	−0.507489
$995$	0	0
$996$	16.0000	0.506979
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	− 16.0000i	− 0.506471i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.n.799.1		2
5.2	odd	4		2850.2.a.ba.1.1		1
5.3	odd	4		570.2.a.c.1.1	✓	1
5.4	even	2	inner	2850.2.d.n.799.2		2
15.2	even	4		8550.2.a.o.1.1		1
15.8	even	4		1710.2.a.n.1.1		1
20.3	even	4		4560.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.c.1.1	✓	1	5.3	odd	4
1710.2.a.n.1.1		1	15.8	even	4
2850.2.a.ba.1.1		1	5.2	odd	4
2850.2.d.n.799.1		2	1.1	even	1	trivial
2850.2.d.n.799.2		2	5.4	even	2	inner
4560.2.a.bd.1.1		1	20.3	even	4
8550.2.a.o.1.1		1	15.2	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} +2.00000i q^{28} -1.00000i q^{32} -2.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} +2.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +8.00000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +2.00000i q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +8.00000i q^{82} +16.0000i q^{83} -2.00000 q^{84} +6.00000 q^{86} -2.00000i q^{88} -16.0000 q^{89} -8.00000i q^{92} -8.00000 q^{94} +1.00000 q^{96} +8.00000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} - 4 q^{34} + 2 q^{36} - 16 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{49} + 4 q^{51} - 2 q^{54} + 4 q^{56} + 16 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} + 16 q^{71} + 8 q^{74} + 2 q^{76} + 16 q^{79} + 2 q^{81} - 4 q^{84} + 12 q^{86} - 32 q^{89} - 16 q^{94} + 2 q^{96} + 4 q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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