LMFDB - Embedded newform 2850.2.d.c.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

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:	$22.7573645761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
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.c.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -8.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -1.00000i q^{38} -6.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +8.00000i q^{43} +8.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -2.00000i q^{87} -6.00000 q^{89} -24.0000 q^{91} +8.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} + 2 q^{24} - 12 q^{26} - 4 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} - 12 q^{39} + 12 q^{41} + 16 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} + 8 q^{56} + 24 q^{59} - 4 q^{61} - 2 q^{64} + 16 q^{69} - 16 q^{71} - 20 q^{74} + 2 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{84} - 16 q^{86} - 12 q^{89} - 48 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 + 2 * q^24 - 12 * q^26 - 4 * q^29 - 16 * q^31 - 4 * q^34 + 2 * q^36 - 12 * q^39 + 12 * q^41 + 16 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^56 + 24 * q^59 - 4 * q^61 - 2 * q^64 + 16 * q^69 - 16 * q^71 - 20 * q^74 + 2 * q^76 + 32 * q^79 + 2 * q^81 + 8 * q^84 - 16 * q^86 - 12 * q^89 - 48 * q^91 - 2 * q^96

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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−4.00000	−1.06904
$15$	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
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	− 1.00000i	− 0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	− 1.00000i	− 0.162221i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 4.00000i	− 0.617213i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	8.00000	1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 6.00000i	− 0.832050i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	− 1.00000i	− 0.132453i
$58$	− 2.00000i	− 0.262613i
$59$	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	$0.540868\pi$
$62$	− 8.00000i	− 1.01600i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 2.00000i	− 0.242536i
$69$	8.00000	0.963087
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	− 6.00000i	− 0.679366i
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−8.00000	−0.862662
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−24.0000	−2.51588
$92$	8.00000i	0.834058i
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	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
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	4.00000i	0.377964i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	2.00000	0.185695
$117$	− 6.00000i	− 0.554700i
$118$	12.0000i	1.10469i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 2.00000i	− 0.181071i
$123$	6.00000i	0.541002i
$124$	8.00000	0.718421
$125$	0	0
$126$	4.00000	0.356348
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	8.00000i	0.681005i
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 9.00000i	− 0.742307i
$148$	− 10.0000i	− 0.821995i
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	1.00000i	0.0811107i
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	16.0000i	1.27289i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	32.0000	2.52195
$162$	1.00000i	0.0785674i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	4.00000i	0.308607i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 8.00000i	− 0.609994i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	− 6.00000i	− 0.449719i
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	− 24.0000i	− 1.77900i
$183$	− 2.00000i	− 0.147844i
$184$	−8.00000	−0.589768
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	− 18.0000i	− 1.26648i
$203$	− 8.00000i	− 0.561490i
$204$	2.00000	0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	8.00000i	0.556038i
$208$	6.00000i	0.416025i
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 2.00000i	− 0.137361i
$213$	− 8.00000i	− 0.548151i
$214$	−20.0000	−1.36717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 32.0000i	− 2.17230i
$218$	− 6.00000i	− 0.406371i
$219$	2.00000	0.135147
$220$	0	0
$221$	−12.0000	−0.807207
$222$	− 10.0000i	− 0.671156i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	18.0000	1.19734
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	1.00000i	0.0662266i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−12.0000	−0.781133
$237$	16.0000i	1.03931i
$238$	− 8.00000i	− 0.518563i
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	− 6.00000i	− 0.381771i
$248$	8.00000i	0.508001i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 8.00000i	− 0.498058i
$259$	−40.0000	−2.48548
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	− 6.00000i	− 0.367194i
$268$	12.0000i	0.733017i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	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
$273$	− 24.0000i	− 1.45255i
$274$	6.00000	0.362473
$275$	0	0
$276$	−8.00000	−0.481543
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	12.0000i	0.719712i
$279$	8.00000	0.478947
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	24.0000i	1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	2.00000i	0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	2.00000i	0.115857i
$299$	48.0000	2.77591
$300$	0	0
$301$	−32.0000	−1.84445
$302$	0	0
$303$	− 18.0000i	− 1.03407i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	6.00000i	0.339683i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	− 2.00000i	− 0.112154i
$319$	0	0
$320$	0	0
$321$	−20.0000	−1.11629
$322$	32.0000i	1.78329i
$323$	− 2.00000i	− 0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	− 6.00000i	− 0.331801i
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 10.0000i	− 0.547997i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−4.00000	−0.218218
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 23.0000i	− 1.25104i
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	1.00000i	0.0540738i
$343$	− 8.00000i	− 0.431959i
$344$	8.00000	0.431331
$345$	0	0
$346$	14.0000	0.752645
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	2.00000i	0.107211i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	6.00000	0.317999
$357$	− 8.00000i	− 0.423405i
$358$	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
$363$	− 11.0000i	− 0.577350i
$364$	24.0000	1.25794
$365$	0	0
$366$	2.00000	0.104542
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	− 8.00000i	− 0.417029i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−8.00000	−0.415339
$372$	8.00000i	0.414781i
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	4.00000i	0.205738i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	− 12.0000i	− 0.613973i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	− 8.00000i	− 0.406663i
$388$	10.0000i	0.507673i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	9.00000i	0.454569i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	38.0000i	1.90717i	0.301131	+	0.953583i	$0.402636\pi$
$397$	38.0000i	1.90717i	−0.301131	+	0.953583i	$0.597364\pi$
$398$	− 16.0000i	− 0.802008i
$399$	4.00000	0.200250
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	12.0000i	0.598506i
$403$	− 48.0000i	− 2.39105i
$404$	18.0000	0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	0	0
$408$	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
$411$	6.00000	0.295958
$412$	8.00000i	0.394132i
$413$	48.0000i	2.36193i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	−6.00000	−0.294174
$417$	12.0000i	0.587643i
$418$	0	0
$419$	40.0000	1.95413	0.977064	−	0.212946i	$-0.0683059\pi$
$419$	40.0000	1.95413	0.977064	+	0.212946i	$0.0683059\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	2.00000	0.0971286
$425$	0	0
$426$	8.00000	0.387601
$427$	− 8.00000i	− 0.387147i
$428$	− 20.0000i	− 0.966736i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	32.0000	1.53605
$435$	0	0
$436$	6.00000	0.287348
$437$	8.00000i	0.382692i
$438$	2.00000i	0.0955637i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	− 12.0000i	− 0.570782i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	−16.0000	−0.757622
$447$	2.00000i	0.0945968i
$448$	− 4.00000i	− 0.188982i
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	18.0000i	0.846649i
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 6.00000i	− 0.280362i
$459$	2.00000	0.0933520
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	6.00000i	0.277350i
$469$	48.0000	2.21643
$470$	0	0
$471$	10.0000	0.460776
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	−16.0000	−0.734904
$475$	0	0
$476$	8.00000	0.366679
$477$	− 2.00000i	− 0.0915737i
$478$	20.0000i	0.914779i
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	− 14.0000i	− 0.637683i
$483$	32.0000i	1.45605i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	2.00000i	0.0905357i
$489$	8.00000	0.361773
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 4.00000i	− 0.180151i
$494$	6.00000	0.269953
$495$	0	0
$496$	−8.00000	−0.359211
$497$	− 32.0000i	− 1.43540i
$498$	− 4.00000i	− 0.179244i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	8.00000i	0.354943i
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000i	0.0441942i
$513$	1.00000i	0.0441511i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	− 40.0000i	− 1.75750i
$519$	14.0000	0.614532
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	2.00000i	0.0875376i
$523$	44.0000i	1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$	44.0000i	1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−12.0000	−0.520756
$532$	4.00000i	0.173422i
$533$	36.0000i	1.55933i
$534$	6.00000	0.259645
$535$	0	0
$536$	−12.0000	−0.518321
$537$	20.0000i	0.863064i
$538$	− 18.0000i	− 0.776035i
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	8.00000i	0.343629i
$543$	− 18.0000i	− 0.772454i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	24.0000	1.02711
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	6.00000i	0.256307i
$549$	2.00000	0.0853579
$550$	0	0
$551$	2.00000	0.0852029
$552$	− 8.00000i	− 0.340503i
$553$	64.0000i	2.72156i
$554$	2.00000	0.0849719
$555$	0	0
$556$	−12.0000	−0.508913
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	8.00000i	0.338667i
$559$	−48.0000	−2.03018
$560$	0	0
$561$	0	0
$562$	− 2.00000i	− 0.0843649i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	−16.0000	−0.672530
$567$	4.00000i	0.167984i
$568$	8.00000i	0.335673i
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	13.0000i	0.540729i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	−16.0000	−0.663792
$582$	10.0000i	0.414513i
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	9.00000i	0.371154i
$589$	8.00000	0.329634
$590$	0	0
$591$	−6.00000	−0.246807
$592$	10.0000i	0.410997i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	− 16.0000i	− 0.654836i
$598$	48.0000i	1.96287i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	− 32.0000i	− 1.30422i
$603$	12.0000i	0.488678i
$604$	0	0
$605$	0	0
$606$	18.0000	0.731200
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	8.00000	0.324176
$610$	0	0
$611$	0	0
$612$	2.00000i	0.0808452i
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	26.0000i	1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$	26.0000i	1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$	8.00000i	0.321807i
$619$	−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
$621$	−8.00000	−0.321029
$622$	12.0000i	0.481156i
$623$	− 24.0000i	− 0.961540i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	10.0000i	0.399043i
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	− 16.0000i	− 0.636446i
$633$	20.0000i	0.794929i
$634$	26.0000	1.03259
$635$	0	0
$636$	2.00000	0.0793052
$637$	− 54.0000i	− 2.13956i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	− 20.0000i	− 0.789337i
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	−32.0000	−1.26098
$645$	0	0
$646$	2.00000	0.0786889
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	32.0000	1.25418
$652$	8.00000i	0.313304i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	6.00000	0.234261
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	20.0000i	0.777322i
$663$	− 12.0000i	− 0.466041i
$664$	4.00000	0.155230
$665$	0	0
$666$	10.0000	0.387492
$667$	16.0000i	0.619522i
$668$	− 8.00000i	− 0.309529i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	− 4.00000i	− 0.154303i
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	23.0000	0.884615
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	18.0000i	0.691286i
$679$	40.0000	1.53506
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	8.00000	0.305441
$687$	− 6.00000i	− 0.228914i
$688$	8.00000i	0.304997i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	12.0000i	0.454532i
$698$	− 14.0000i	− 0.529908i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	6.00000i	0.226455i
$703$	− 10.0000i	− 0.377157i
$704$	0	0
$705$	0	0
$706$	2.00000	0.0752710
$707$	− 72.0000i	− 2.70784i
$708$	− 12.0000i	− 0.450988i
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	6.00000i	0.224860i
$713$	64.0000i	2.39682i
$714$	8.00000	0.299392
$715$	0	0
$716$	−20.0000	−0.747435
$717$	20.0000i	0.746914i
$718$	− 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
$721$	32.0000	1.19174
$722$	1.00000i	0.0372161i
$723$	− 14.0000i	− 0.520666i
$724$	18.0000	0.668965
$725$	0	0
$726$	11.0000	0.408248
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	24.0000i	0.889499i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	2.00000i	0.0739221i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	8.00000	0.294884
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	− 8.00000i	− 0.293689i
$743$	32.0000i	1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$	32.0000i	1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	34.0000	1.24483
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	−80.0000	−2.92314
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	−4.00000	−0.145479
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	8.00000i	0.289809i
$763$	− 24.0000i	− 0.868858i
$764$	12.0000	0.434145
$765$	0	0
$766$	−24.0000	−0.867155
$767$	72.0000i	2.59977i
$768$	1.00000i	0.0360844i
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	− 2.00000i	− 0.0719816i
$773$	− 46.0000i	− 1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$	− 46.0000i	− 1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−10.0000	−0.358979
$777$	− 40.0000i	− 1.43499i
$778$	− 6.00000i	− 0.215110i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	16.0000i	0.572159i
$783$	2.00000i	0.0714742i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	72.0000	2.56003
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	16.0000	0.567105
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	6.00000i	0.211867i
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	48.0000	1.69073
$807$	− 18.0000i	− 0.633630i
$808$	18.0000i	0.633238i
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	8.00000i	0.280745i
$813$	8.00000i	0.280572i
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 8.00000i	− 0.279885i
$818$	− 18.0000i	− 0.629355i
$819$	24.0000	0.838628
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	6.00000i	0.209274i
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−48.0000	−1.67013
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	− 8.00000i	− 0.278019i
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	− 6.00000i	− 0.208013i
$833$	− 18.0000i	− 0.623663i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	40.0000i	1.38178i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	22.0000i	0.758170i
$843$	− 2.00000i	− 0.0688837i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	2.00000i	0.0686803i
$849$	−16.0000	−0.549119
$850$	0	0
$851$	80.0000	2.74236
$852$	8.00000i	0.274075i
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	20.0000	0.683586
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	24.0000i	0.817443i
$863$	− 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
$868$	32.0000i	1.08615i
$869$	0	0
$870$	0	0
$871$	72.0000	2.43963
$872$	6.00000i	0.203186i
$873$	10.0000i	0.338449i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	− 8.00000i	− 0.269987i
$879$	6.00000	0.202375
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	9.00000i	0.303046i
$883$	− 8.00000i	− 0.269221i	−0.990899	−	0.134611i	$-0.957022\pi$
$883$	− 8.00000i	− 0.269221i	0.990899	−	0.134611i	$-0.0429784\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 16.0000i	− 0.537227i	−0.963248	−	0.268614i	$-0.913434\pi$
$887$	− 16.0000i	− 0.537227i	0.963248	−	0.268614i	$-0.0865655\pi$
$888$	10.0000i	0.335578i
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	4.00000	0.133631
$897$	48.0000i	1.60267i
$898$	− 30.0000i	− 1.00111i
$899$	16.0000	0.533630
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 32.0000i	− 1.06489i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	0	0
$907$	− 60.0000i	− 1.99227i	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	− 60.0000i	− 1.99227i	0.0878507	−	0.996134i	$-0.472000\pi$
$908$	− 20.0000i	− 0.663723i
$909$	18.0000	0.597022
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	6.00000	0.198246
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	6.00000i	0.197599i
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	8.00000i	0.262754i
$928$	− 2.00000i	− 0.0656532i
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	− 6.00000i	− 0.196537i
$933$	12.0000i	0.392862i
$934$	28.0000	0.916188
$935$	0	0
$936$	−6.00000	−0.196116
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	48.0000i	1.56726i
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	10.0000i	0.325818i
$943$	− 48.0000i	− 1.56310i
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	− 16.0000i	− 0.519656i
$949$	12.0000	0.389536
$950$	0	0
$951$	26.0000	0.843108
$952$	8.00000i	0.259281i
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−20.0000	−0.646846
$957$	0	0
$958$	− 36.0000i	− 1.16311i
$959$	24.0000	0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	− 60.0000i	− 1.93448i
$963$	− 20.0000i	− 0.644491i
$964$	14.0000	0.450910
$965$	0	0
$966$	−32.0000	−1.02958
$967$	44.0000i	1.41494i	0.706741	+	0.707472i	$0.250165\pi$
$967$	44.0000i	1.41494i	−0.706741	+	0.707472i	$0.749835\pi$
$968$	11.0000i	0.353553i
$969$	2.00000	0.0642493
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	48.0000i	1.53881i
$974$	8.00000	0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	8.00000i	0.255812i
$979$	0	0
$980$	0	0
$981$	6.00000	0.191565
$982$	− 40.0000i	− 1.27645i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	6.00000i	0.190885i
$989$	64.0000	2.03508
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	− 8.00000i	− 0.254000i
$993$	20.0000i	0.634681i
$994$	32.0000	1.01498
$995$	0	0
$996$	4.00000	0.126745
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	20.0000i	0.633089i
$999$	10.0000	0.316386

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -8.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -1.00000i q^{38} -6.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +8.00000i q^{43} +8.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -2.00000i q^{87} -6.00000 q^{89} -24.0000 q^{91} +8.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} + 2 q^{24} - 12 q^{26} - 4 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} - 12 q^{39} + 12 q^{41} + 16 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} + 8 q^{56} + 24 q^{59} - 4 q^{61} - 2 q^{64} + 16 q^{69} - 16 q^{71} - 20 q^{74} + 2 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{84} - 16 q^{86} - 12 q^{89} - 48 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 + 2 * q^24 - 12 * q^26 - 4 * q^29 - 16 * q^31 - 4 * q^34 + 2 * q^36 - 12 * q^39 + 12 * q^41 + 16 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^56 + 24 * q^59 - 4 * q^61 - 2 * q^64 + 16 * q^69 - 16 * q^71 - 20 * q^74 + 2 * q^76 + 32 * q^79 + 2 * q^81 + 8 * q^84 - 16 * q^86 - 12 * q^89 - 48 * q^91 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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