LMFDB - Embedded newform 2850.2.d.i.799.2

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

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +1.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +1.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +6.00000 q^{74} -1.00000 q^{76} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} +4.00000i q^{92} +4.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 2 q^{19} + 2 q^{24} + 4 q^{26} - 12 q^{29} + 8 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} + 8 q^{69} + 16 q^{71} + 12 q^{74} - 2 q^{76} + 8 q^{79} + 2 q^{81} - 8 q^{86} - 20 q^{89} + 24 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 2 * q^19 + 2 * q^24 + 4 * q^26 - 12 * q^29 + 8 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 - 8 * q^44 + 8 * q^46 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 24 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 + 8 * q^69 + 16 * q^71 + 12 * q^74 - 2 * q^76 + 8 * q^79 + 2 * q^81 - 8 * q^86 - 20 * q^89 + 24 * q^94 - 2 * q^96 - 8 * q^99

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	4.00000i	0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	1.00000i	0.162221i
$39$	2.00000	0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	4.00000	0.589768
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	2.00000i	0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	− 6.00000i	− 0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 2.00000i	− 0.242536i
$69$	4.00000	0.481543
$70$	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
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	2.00000i	0.226455i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000i	1.10432i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 6.00000i	− 0.643268i
$88$	− 4.00000i	− 0.426401i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	4.00000i	0.417029i
$93$	4.00000i	0.414781i
$94$	12.0000	1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	7.00000i	0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	− 2.00000i	− 0.198030i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	1.00000i	0.0962250i
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	10.0000i	0.901670i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	4.00000i	0.340503i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	8.00000i	0.671345i
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	7.00000i	0.577350i
$148$	6.00000i	0.493197i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	4.00000i	0.318223i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 4.00000i	− 0.304997i
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	12.0000i	0.901975i
$178$	− 10.0000i	− 0.749532i
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−4.00000	−0.293294
$187$	8.00000i	0.585018i
$188$	12.0000i	0.875190i
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 4.00000i	− 0.284268i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	10.0000i	0.703598i
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	−4.00000	−0.278693
$207$	4.00000i	0.278019i
$208$	− 2.00000i	− 0.138675i
$209$	4.00000	0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	8.00000i	0.548151i
$214$	4.00000	0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000i	0.406371i
$219$	−6.00000	−0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	6.00000i	0.402694i
$223$	− 12.0000i	− 0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$	− 12.0000i	− 0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$	0	0
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	4.00000i	0.259828i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	5.00000i	0.321412i
$243$	1.00000i	0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	−10.0000	−0.637577
$247$	− 2.00000i	− 0.127257i
$248$	− 4.00000i	− 0.254000i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	− 12.0000i	− 0.741362i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	− 4.00000i	− 0.244339i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−4.00000	−0.240772
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	20.0000i	1.19952i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	12.0000i	0.714590i
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 6.00000i	− 0.351123i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	− 4.00000i	− 0.232104i
$298$	− 18.0000i	− 1.04271i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	0	0
$302$	− 12.0000i	− 0.690522i
$303$	10.0000i	0.574485i
$304$	1.00000	0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	− 2.00000i	− 0.113228i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	6.00000i	0.336463i
$319$	−24.0000	−1.34374
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	2.00000i	0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	6.00000i	0.331801i
$328$	− 10.0000i	− 0.552158i
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	− 12.0000i	− 0.658586i
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	9.00000i	0.489535i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	16.0000	0.866449
$342$	− 1.00000i	− 0.0540738i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−10.0000	−0.537603
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	6.00000i	0.321634i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	4.00000i	0.213201i
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	− 20.0000i	− 1.05703i
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.0000i	0.946059i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	− 4.00000i	− 0.207390i
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−12.0000	−0.618853
$377$	12.0000i	0.618031i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	20.0000i	1.02329i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	− 4.00000i	− 0.203331i
$388$	− 2.00000i	− 0.101535i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	− 7.00000i	− 0.353553i
$393$	− 12.0000i	− 0.605320i
$394$	−2.00000	−0.100759
$395$	0	0
$396$	4.00000	0.201008
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 8.00000i	− 0.398508i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	0	0
$407$	− 24.0000i	− 1.18964i
$408$	2.00000i	0.0990148i
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	− 4.00000i	− 0.197066i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	20.0000i	0.979404i
$418$	4.00000i	0.195646i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	− 4.00000i	− 0.194717i
$423$	12.0000i	0.583460i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	0	0
$428$	4.00000i	0.193347i
$429$	8.00000	0.386244
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 4.00000i	− 0.191346i
$438$	− 6.00000i	− 0.286691i
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	4.00000i	0.190261i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	12.0000	0.568216
$447$	− 18.0000i	− 0.851371i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	− 14.0000i	− 0.658505i
$453$	− 12.0000i	− 0.563809i
$454$	20.0000	0.938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	10.0000i	0.467269i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	− 12.0000i	− 0.552345i
$473$	16.0000i	0.735681i
$474$	−4.00000	−0.183726
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	− 12.0000i	− 0.548867i
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	26.0000i	1.18427i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	2.00000i	0.0905357i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 12.0000i	− 0.540453i
$494$	2.00000	0.0899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	− 20.0000i	− 0.892644i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	16.0000	0.711287
$507$	9.00000i	0.399704i
$508$	4.00000i	0.177471i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	4.00000	0.176090
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000i	0.262613i
$523$	− 28.0000i	− 1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$	− 28.0000i	− 1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−4.00000	−0.174408
$527$	8.00000i	0.348485i
$528$	4.00000i	0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	10.0000	0.432742
$535$	0	0
$536$	4.00000	0.172774
$537$	− 20.0000i	− 0.863064i
$538$	2.00000i	0.0862261i
$539$	28.0000	1.20605
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	8.00000i	0.343629i
$543$	18.0000i	0.772454i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 10.0000i	− 0.427179i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−6.00000	−0.255609
$552$	− 4.00000i	− 0.170251i
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−20.0000	−0.848189
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	− 4.00000i	− 0.169334i
$559$	8.00000	0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	− 6.00000i	− 0.253095i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−28.0000	−1.17693
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	8.00000i	0.334497i
$573$	20.0000i	0.835512i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	13.0000i	0.540729i
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	− 2.00000i	− 0.0829027i
$583$	− 24.0000i	− 0.993978i
$584$	6.00000	0.248282
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	− 7.00000i	− 0.288675i
$589$	4.00000	0.164817
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	− 6.00000i	− 0.246598i
$593$	− 10.0000i	− 0.410651i	−0.978694	−	0.205325i	$-0.934175\pi$
$593$	− 10.0000i	− 0.410651i	0.978694	−	0.205325i	$-0.0658253\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	− 8.00000i	− 0.327144i
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	12.0000	0.488273
$605$	0	0
$606$	−10.0000	−0.406222
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	1.00000i	0.0405554i
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	2.00000i	0.0808452i
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 14.0000i	− 0.563619i	−0.959470	−	0.281809i	$-0.909065\pi$
$617$	− 14.0000i	− 0.563619i	0.959470	−	0.281809i	$-0.0909346\pi$
$618$	− 4.00000i	− 0.160904i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 28.0000i	− 1.12270i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	−6.00000	−0.239808
$627$	4.00000i	0.159745i
$628$	2.00000i	0.0798087i
$629$	12.0000	0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	− 4.00000i	− 0.159111i
$633$	− 4.00000i	− 0.158986i
$634$	26.0000	1.03259
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 14.0000i	− 0.554700i
$638$	− 24.0000i	− 0.950169i
$639$	−8.00000	−0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000i	0.157867i
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	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
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	10.0000	0.390434
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	28.0000i	1.08825i
$663$	4.00000i	0.155347i
$664$	12.0000	0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000i	0.929284i
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	− 14.0000i	− 0.537667i
$679$	0	0
$680$	0	0
$681$	20.0000	0.766402
$682$	16.0000i	0.612672i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	4.00000i	0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	− 10.0000i	− 0.380143i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	20.0000i	0.757554i
$698$	10.0000i	0.378506i
$699$	18.0000	0.680823
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 6.00000i	− 0.226294i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	26.0000	0.978523
$707$	0	0
$708$	− 12.0000i	− 0.450988i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	10.0000i	0.374766i
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	− 12.0000i	− 0.448148i
$718$	− 12.0000i	− 0.447836i
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	0	0
$721$	0	0
$722$	1.00000i	0.0372161i
$723$	26.0000i	0.966950i
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−5.00000	−0.185567
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	2.00000i	0.0739221i
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	4.00000	0.147442
$737$	16.0000i	0.589368i
$738$	− 10.0000i	− 0.368105i
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−22.0000	−0.805477
$747$	− 12.0000i	− 0.439057i
$748$	− 8.00000i	− 0.292509i
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 12.0000i	− 0.437595i
$753$	− 20.0000i	− 0.728841i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	20.0000i	0.726433i
$759$	16.0000	0.580763
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	4.00000i	0.144905i
$763$	0	0
$764$	−20.0000	−0.723575
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	1.00000i	0.0360844i
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	2.00000i	0.0719816i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	22.0000i	0.788738i
$779$	10.0000	0.358287
$780$	0	0
$781$	32.0000	1.14505
$782$	8.00000i	0.286079i
$783$	6.00000i	0.214423i
$784$	7.00000	0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	−4.00000	−0.142404
$790$	0	0
$791$	0	0
$792$	4.00000i	0.142134i
$793$	4.00000i	0.142044i
$794$	18.0000	0.638796
$795$	0	0
$796$	0	0
$797$	54.0000i	1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$	54.0000i	1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	10.0000	0.353333
$802$	− 14.0000i	− 0.494357i
$803$	24.0000i	0.846942i
$804$	4.00000	0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	2.00000i	0.0704033i
$808$	− 10.0000i	− 0.351799i
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	24.0000	0.841200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	4.00000i	0.139942i
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	− 10.0000i	− 0.348790i
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	2.00000i	0.0693375i
$833$	14.0000i	0.485071i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	−4.00000	−0.138343
$837$	− 4.00000i	− 0.138260i
$838$	− 12.0000i	− 0.414533i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000i	0.896019i
$843$	− 6.00000i	− 0.206651i
$844$	4.00000	0.137686
$845$	0	0
$846$	−12.0000	−0.412568
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−24.0000	−0.822709
$852$	− 8.00000i	− 0.274075i
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	26.0000i	0.888143i	0.895991	+	0.444072i	$0.146466\pi$
$857$	26.0000i	0.888143i	−0.895991	+	0.444072i	$0.853534\pi$
$858$	8.00000i	0.273115i
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	8.00000i	0.272481i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	34.0000	1.15537
$867$	13.0000i	0.441503i
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	8.00000	0.271070
$872$	− 6.00000i	− 0.203186i
$873$	− 2.00000i	− 0.0676897i
$874$	4.00000	0.135302
$875$	0	0
$876$	6.00000	0.202721
$877$	− 46.0000i	− 1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$	− 46.0000i	− 1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$	− 12.0000i	− 0.404980i
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	− 7.00000i	− 0.235702i
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	4.00000	0.134383
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	12.0000i	0.401790i
$893$	− 12.0000i	− 0.401565i
$894$	18.0000	0.602010
$895$	0	0
$896$	0	0
$897$	− 8.00000i	− 0.267112i
$898$	30.0000i	1.00111i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	12.0000	0.399778
$902$	40.0000i	1.33185i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	12.0000	0.398673
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	20.0000i	0.663723i
$909$	−10.0000	−0.331679
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	1.00000i	0.0331133i
$913$	48.0000i	1.58857i
$914$	6.00000	0.198462
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	− 38.0000i	− 1.25146i
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	− 4.00000i	− 0.131377i
$928$	− 6.00000i	− 0.196960i
$929$	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
$931$	7.00000	0.229416
$932$	18.0000i	0.589610i
$933$	− 28.0000i	− 0.916679i
$934$	28.0000	0.916188
$935$	0	0
$936$	2.00000	0.0653720
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	2.00000i	0.0651635i
$943$	− 40.0000i	− 1.30258i
$944$	12.0000	0.390567
$945$	0	0
$946$	−16.0000	−0.520205
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	− 4.00000i	− 0.129914i
$949$	12.0000	0.389536
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	− 42.0000i	− 1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$	− 42.0000i	− 1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	12.0000	0.388108
$957$	− 24.0000i	− 0.775810i
$958$	− 20.0000i	− 0.646171i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 12.0000i	− 0.386896i
$963$	4.00000i	0.128898i
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 5.00000i	− 0.160706i
$969$	−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
$973$	0	0
$974$	20.0000	0.640841
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	− 4.00000i	− 0.127906i
$979$	−40.0000	−1.27841
$980$	0	0
$981$	−6.00000	−0.191565
$982$	− 36.0000i	− 1.14881i
$983$	− 56.0000i	− 1.78612i	−0.449935	−	0.893061i	$-0.648553\pi$
$983$	− 56.0000i	− 1.78612i	0.449935	−	0.893061i	$-0.351447\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	2.00000i	0.0636285i
$989$	16.0000	0.508770
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	4.00000i	0.127000i
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	38.0000i	1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$	38.0000i	1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$	36.0000i	1.13956i
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.i.799.2		2
5.2	odd	4		2850.2.a.m.1.1		1
5.3	odd	4		570.2.a.g.1.1	✓	1
5.4	even	2	inner	2850.2.d.i.799.1		2
15.2	even	4		8550.2.a.x.1.1		1
15.8	even	4		1710.2.a.i.1.1		1
20.3	even	4		4560.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.g.1.1	✓	1	5.3	odd	4
1710.2.a.i.1.1		1	15.8	even	4
2850.2.a.m.1.1		1	5.2	odd	4
2850.2.d.i.799.1		2	5.4	even	2	inner
2850.2.d.i.799.2		2	1.1	even	1	trivial
4560.2.a.s.1.1		1	20.3	even	4
8550.2.a.x.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +1.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +1.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +6.00000 q^{74} -1.00000 q^{76} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} +4.00000i q^{92} +4.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 2 q^{19} + 2 q^{24} + 4 q^{26} - 12 q^{29} + 8 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} + 8 q^{69} + 16 q^{71} + 12 q^{74} - 2 q^{76} + 8 q^{79} + 2 q^{81} - 8 q^{86} - 20 q^{89} + 24 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 2 * q^19 + 2 * q^24 + 4 * q^26 - 12 * q^29 + 8 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 - 8 * q^44 + 8 * q^46 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 24 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 + 8 * q^69 + 16 * q^71 + 12 * q^74 - 2 * q^76 + 8 * q^79 + 2 * q^81 - 8 * q^86 - 20 * q^89 + 24 * q^94 - 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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