LMFDB - Embedded newform 1850.2.b.g.149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.7723243739$
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 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.g.149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +3.00000i q^{22} +6.00000i q^{23} -2.00000 q^{24} +4.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -2.00000i q^{38} -8.00000 q^{39} +3.00000 q^{41} +2.00000i q^{42} -1.00000i q^{43} -3.00000 q^{44} -6.00000 q^{46} -12.0000i q^{47} -2.00000i q^{48} +6.00000 q^{49} -6.00000 q^{51} +4.00000i q^{52} +3.00000i q^{53} +4.00000 q^{54} +1.00000 q^{56} +4.00000i q^{57} -3.00000i q^{58} -1.00000 q^{61} +5.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +4.00000i q^{67} +3.00000i q^{68} +12.0000 q^{69} +6.00000 q^{71} +1.00000i q^{72} -16.0000i q^{73} +1.00000 q^{74} +2.00000 q^{76} +3.00000i q^{77} -8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} +3.00000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +1.00000 q^{86} +6.00000i q^{87} -3.00000i q^{88} +6.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} -10.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} -17.0000i q^{97} +6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 6 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} - 6 q^{29} + 10 q^{31} + 6 q^{34} + 2 q^{36} - 16 q^{39} + 6 q^{41} - 6 q^{44} - 12 q^{46} + 12 q^{49} - 12 q^{51} + 8 q^{54} + 2 q^{56} - 2 q^{61} - 2 q^{64} + 12 q^{66} + 24 q^{69} + 12 q^{71} + 2 q^{74} + 4 q^{76} - 16 q^{79} - 22 q^{81} - 4 q^{84} + 2 q^{86} + 12 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 6 * q^11 - 2 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^21 - 4 * q^24 + 8 * q^26 - 6 * q^29 + 10 * q^31 + 6 * q^34 + 2 * q^36 - 16 * q^39 + 6 * q^41 - 6 * q^44 - 12 * q^46 + 12 * q^49 - 12 * q^51 + 8 * q^54 + 2 * q^56 - 2 * q^61 - 2 * q^64 + 12 * q^66 + 24 * q^69 + 12 * q^71 + 2 * q^74 + 4 * q^76 - 16 * q^79 - 22 * q^81 - 4 * q^84 + 2 * q^86 + 12 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000i	0.577350i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	3.00000i	0.639602i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	4.00000	0.784465
$27$	− 4.00000i	− 0.769800i
$28$	− 1.00000i	− 0.188982i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000i	0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	− 2.00000i	− 0.324443i
$39$	−8.00000	−1.28103
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	2.00000i	0.308607i
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	− 2.00000i	− 0.288675i
$49$	6.00000	0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	4.00000i	0.554700i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	4.00000i	0.529813i
$58$	− 3.00000i	− 0.393919i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	5.00000i	0.635001i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	3.00000i	0.363803i
$69$	12.0000	1.44463
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000i	0.117851i
$73$	− 16.0000i	− 1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	− 16.0000i	− 1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	3.00000i	0.341882i
$78$	− 8.00000i	− 0.905822i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	3.00000i	0.331295i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	1.00000	0.107833
$87$	6.00000i	0.643268i
$88$	− 3.00000i	− 0.319801i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 6.00000i	− 0.625543i
$93$	− 10.0000i	− 1.03695i
$94$	12.0000	1.23771
$95$	0	0
$96$	2.00000	0.204124
$97$	− 17.0000i	− 1.72609i	−0.505128	−	0.863044i	$-0.668555\pi$
$97$	− 17.0000i	− 1.72609i	0.505128	−	0.863044i	$-0.331445\pi$
$98$	6.00000i	0.606092i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	− 6.00000i	− 0.594089i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−3.00000	−0.291386
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	4.00000i	0.384900i
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000i	0.0944911i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	3.00000	0.278543
$117$	4.00000i	0.369800i
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 1.00000i	− 0.0905357i
$123$	− 6.00000i	− 0.541002i
$124$	−5.00000	−0.449013
$125$	0	0
$126$	1.00000	0.0890871
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	6.00000i	0.522233i
$133$	− 2.00000i	− 0.173422i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	12.0000i	1.02151i
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	6.00000i	0.503509i
$143$	− 12.0000i	− 1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	16.0000	1.32417
$147$	− 12.0000i	− 0.989743i
$148$	1.00000i	0.0821995i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	2.00000i	0.162221i
$153$	3.00000i	0.242536i
$154$	−3.00000	−0.241747
$155$	0	0
$156$	8.00000	0.640513
$157$	13.0000i	1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$	13.0000i	1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$	− 8.00000i	− 0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	−6.00000	−0.472866
$162$	− 11.0000i	− 0.864242i
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	2.00000	0.152944
$172$	1.00000i	0.0762493i
$173$	15.0000i	1.14043i	0.821496	+	0.570214i	$0.193140\pi$
$173$	15.0000i	1.14043i	−0.821496	+	0.570214i	$0.806860\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	6.00000i	0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.00000i	0.296500i
$183$	2.00000i	0.147844i
$184$	6.00000	0.442326
$185$	0	0
$186$	10.0000	0.733236
$187$	− 9.00000i	− 0.658145i
$188$	12.0000i	0.875190i
$189$	4.00000	0.290957
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	2.00000i	0.144338i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	17.0000	1.22053
$195$	0	0
$196$	−6.00000	−0.428571
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 3.00000i	− 0.213201i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 6.00000i	− 0.422159i
$203$	− 3.00000i	− 0.210559i
$204$	6.00000	0.420084
$205$	0	0
$206$	4.00000	0.278693
$207$	− 6.00000i	− 0.417029i
$208$	− 4.00000i	− 0.277350i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	− 3.00000i	− 0.206041i
$213$	− 12.0000i	− 0.822226i
$214$	6.00000	0.410152
$215$	0	0
$216$	−4.00000	−0.272166
$217$	5.00000i	0.339422i
$218$	− 11.0000i	− 0.745014i
$219$	−32.0000	−2.16236
$220$	0	0
$221$	−12.0000	−0.807207
$222$	− 2.00000i	− 0.134231i
$223$	17.0000i	1.13840i	0.822198	+	0.569202i	$0.192748\pi$
$223$	17.0000i	1.13840i	−0.822198	+	0.569202i	$0.807252\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−9.00000	−0.598671
$227$	27.0000i	1.79205i	0.444001	+	0.896026i	$0.353559\pi$
$227$	27.0000i	1.79205i	−0.444001	+	0.896026i	$0.646441\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	3.00000i	0.196960i
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	3.00000i	0.194461i
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	− 2.00000i	− 0.128565i
$243$	10.0000i	0.641500i
$244$	1.00000	0.0640184
$245$	0	0
$246$	6.00000	0.382546
$247$	8.00000i	0.509028i
$248$	− 5.00000i	− 0.317500i
$249$	−24.0000	−1.52094
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	1.00000i	0.0629941i
$253$	18.0000i	1.13165i
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	− 2.00000i	− 0.124515i
$259$	1.00000	0.0621370
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	9.00000i	0.554964i	0.960731	+	0.277482i	$0.0894999\pi$
$263$	9.00000i	0.554964i	−0.960731	+	0.277482i	$0.910500\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	2.00000	0.122628
$267$	− 12.0000i	− 0.734388i
$268$	− 4.00000i	− 0.244339i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	− 3.00000i	− 0.181902i
$273$	− 8.00000i	− 0.484182i
$274$	12.0000	0.724947
$275$	0	0
$276$	−12.0000	−0.722315
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	13.0000i	0.779688i
$279$	−5.00000	−0.299342
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	− 24.0000i	− 1.42918i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	3.00000i	0.177084i
$288$	− 1.00000i	− 0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−34.0000	−1.99312
$292$	16.0000i	0.936329i
$293$	21.0000i	1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$	21.0000i	1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	12.0000	0.699854
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	− 12.0000i	− 0.696311i
$298$	− 6.00000i	− 0.347571i
$299$	24.0000	1.38796
$300$	0	0
$301$	1.00000	0.0576390
$302$	8.00000i	0.460348i
$303$	12.0000i	0.689382i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−3.00000	−0.171499
$307$	34.0000i	1.94048i	0.242140	+	0.970241i	$0.422151\pi$
$307$	34.0000i	1.94048i	−0.242140	+	0.970241i	$0.577849\pi$
$308$	− 3.00000i	− 0.170941i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	8.00000i	0.452911i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	8.00000	0.450035
$317$	21.0000i	1.17948i	0.807594	+	0.589739i	$0.200769\pi$
$317$	21.0000i	1.17948i	−0.807594	+	0.589739i	$0.799231\pi$
$318$	6.00000i	0.336463i
$319$	−9.00000	−0.503903
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 6.00000i	− 0.334367i
$323$	6.00000i	0.333849i
$324$	11.0000	0.611111
$325$	0	0
$326$	−11.0000	−0.609234
$327$	22.0000i	1.21660i
$328$	− 3.00000i	− 0.165647i
$329$	12.0000	0.661581
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	12.0000i	0.658586i
$333$	1.00000i	0.0547997i
$334$	12.0000	0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	4.00000i	0.217894i	0.994048	+	0.108947i	$0.0347479\pi$
$337$	4.00000i	0.217894i	−0.994048	+	0.108947i	$0.965252\pi$
$338$	− 3.00000i	− 0.163178i
$339$	18.0000	0.977626
$340$	0	0
$341$	15.0000	0.812296
$342$	2.00000i	0.108148i
$343$	13.0000i	0.701934i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−15.0000	−0.806405
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	3.00000i	0.159901i
$353$	− 21.0000i	− 1.11772i	−0.829263	−	0.558859i	$-0.811239\pi$
$353$	− 21.0000i	− 1.11772i	0.829263	−	0.558859i	$-0.188761\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 6.00000i	− 0.317554i
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000i	0.105118i
$363$	4.00000i	0.209946i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 35.0000i	− 1.82699i	−0.406855	−	0.913493i	$-0.633375\pi$
$367$	− 35.0000i	− 1.82699i	0.406855	−	0.913493i	$-0.366625\pi$
$368$	6.00000i	0.312772i
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−3.00000	−0.155752
$372$	10.0000i	0.518476i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−12.0000	−0.618853
$377$	12.0000i	0.618031i
$378$	4.00000i	0.205738i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−40.0000	−2.04926
$382$	3.00000i	0.153493i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−14.0000	−0.712581
$387$	1.00000i	0.0508329i
$388$	17.0000i	0.863044i
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	− 6.00000i	− 0.303046i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	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	$0.452133\pi$
$402$	8.00000i	0.399004i
$403$	− 20.0000i	− 0.996271i
$404$	6.00000	0.298511
$405$	0	0
$406$	3.00000	0.148888
$407$	− 3.00000i	− 0.148704i
$408$	6.00000i	0.297044i
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	4.00000i	0.197066i
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	4.00000	0.196116
$417$	− 26.0000i	− 1.27323i
$418$	− 6.00000i	− 0.293470i
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	5.00000i	0.243396i
$423$	12.0000i	0.583460i
$424$	3.00000	0.145693
$425$	0	0
$426$	12.0000	0.581402
$427$	− 1.00000i	− 0.0483934i
$428$	6.00000i	0.290021i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	− 4.00000i	− 0.192450i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	11.0000	0.526804
$437$	− 12.0000i	− 0.574038i
$438$	− 32.0000i	− 1.52902i
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	− 12.0000i	− 0.570782i
$443$	30.0000i	1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$	30.0000i	1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−17.0000	−0.804973
$447$	12.0000i	0.567581i
$448$	− 1.00000i	− 0.0472456i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	− 9.00000i	− 0.423324i
$453$	− 16.0000i	− 0.751746i
$454$	−27.0000	−1.26717
$455$	0	0
$456$	4.00000	0.187317
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−12.0000	−0.560112
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	6.00000i	0.279145i
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	0	0
$467$	− 27.0000i	− 1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$	− 27.0000i	− 1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	− 4.00000i	− 0.184900i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	26.0000	1.19802
$472$	0	0
$473$	− 3.00000i	− 0.137940i
$474$	−16.0000	−0.734904
$475$	0	0
$476$	−3.00000	−0.137505
$477$	− 3.00000i	− 0.137361i
$478$	9.00000i	0.411650i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	− 28.0000i	− 1.27537i
$483$	12.0000i	0.546019i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−10.0000	−0.453609
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	1.00000i	0.0452679i
$489$	22.0000	0.994874
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000i	0.270501i
$493$	9.00000i	0.405340i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	5.00000	0.224507
$497$	6.00000i	0.269137i
$498$	− 24.0000i	− 1.07547i
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	18.0000i	0.803379i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−18.0000	−0.800198
$507$	6.00000i	0.266469i
$508$	20.0000i	0.887357i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000i	0.0441942i
$513$	8.00000i	0.353209i
$514$	18.0000	0.793946
$515$	0	0
$516$	2.00000	0.0880451
$517$	− 36.0000i	− 1.58328i
$518$	1.00000i	0.0439375i
$519$	30.0000	1.31685
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	3.00000i	0.131306i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	− 15.0000i	− 0.653410i
$528$	− 6.00000i	− 0.261116i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	2.00000i	0.0867110i
$533$	− 12.0000i	− 0.519778i
$534$	12.0000	0.519291
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	18.0000i	0.776035i
$539$	18.0000	0.775315
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	2.00000i	0.0859074i
$543$	− 4.00000i	− 0.171656i
$544$	3.00000	0.128624
$545$	0	0
$546$	8.00000	0.342368
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	12.0000i	0.512615i
$549$	1.00000	0.0426790
$550$	0	0
$551$	6.00000	0.255609
$552$	− 12.0000i	− 0.510754i
$553$	− 8.00000i	− 0.340195i
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−13.0000	−0.551323
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	− 5.00000i	− 0.211667i
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−18.0000	−0.759961
$562$	24.0000i	1.01238i
$563$	− 3.00000i	− 0.126435i	−0.998000	−	0.0632175i	$-0.979864\pi$
$563$	− 3.00000i	− 0.126435i	0.998000	−	0.0632175i	$-0.0201362\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	4.00000	0.168133
$567$	− 11.0000i	− 0.461957i
$568$	− 6.00000i	− 0.251754i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	12.0000i	0.501745i
$573$	− 6.00000i	− 0.250654i
$574$	−3.00000	−0.125218
$575$	0	0
$576$	1.00000	0.0416667
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	8.00000i	0.332756i
$579$	28.0000	1.16364
$580$	0	0
$581$	12.0000	0.497844
$582$	− 34.0000i	− 1.40935i
$583$	9.00000i	0.372742i
$584$	−16.0000	−0.662085
$585$	0	0
$586$	−21.0000	−0.867502
$587$	− 27.0000i	− 1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$587$	− 27.0000i	− 1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$588$	12.0000i	0.494872i
$589$	−10.0000	−0.412043
$590$	0	0
$591$	12.0000	0.493614
$592$	− 1.00000i	− 0.0410997i
$593$	− 36.0000i	− 1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$	− 36.0000i	− 1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	6.00000	0.245770
$597$	− 32.0000i	− 1.30967i
$598$	24.0000i	0.981433i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	1.00000i	0.0407570i
$603$	− 4.00000i	− 0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−12.0000	−0.487467
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−48.0000	−1.94187
$612$	− 3.00000i	− 0.121268i
$613$	29.0000i	1.17130i	0.810564	+	0.585649i	$0.199160\pi$
$613$	29.0000i	1.17130i	−0.810564	+	0.585649i	$0.800840\pi$
$614$	−34.0000	−1.37213
$615$	0	0
$616$	3.00000	0.120873
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	− 9.00000i	− 0.360867i
$623$	6.00000i	0.240385i
$624$	−8.00000	−0.320256
$625$	0	0
$626$	−26.0000	−1.03917
$627$	12.0000i	0.479234i
$628$	− 13.0000i	− 0.518756i
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	8.00000i	0.318223i
$633$	− 10.0000i	− 0.397464i
$634$	−21.0000	−0.834017
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 24.0000i	− 0.950915i
$638$	− 9.00000i	− 0.356313i
$639$	−6.00000	−0.237356
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 13.0000i	− 0.512670i	−0.966588	−	0.256335i	$-0.917485\pi$
$643$	− 13.0000i	− 0.512670i	0.966588	−	0.256335i	$-0.0825150\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	−6.00000	−0.236067
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	11.0000i	0.432121i
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	− 11.0000i	− 0.430793i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−22.0000	−0.860268
$655$	0	0
$656$	3.00000	0.117130
$657$	16.0000i	0.624219i
$658$	12.0000i	0.467809i
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	26.0000i	1.01052i
$663$	24.0000i	0.932083i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	− 18.0000i	− 0.696963i
$668$	12.0000i	0.464294i
$669$	34.0000	1.31452
$670$	0	0
$671$	−3.00000	−0.115814
$672$	2.00000i	0.0771517i
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	3.00000	0.115385
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	18.0000i	0.691286i
$679$	17.0000	0.652400
$680$	0	0
$681$	54.0000	2.06928
$682$	15.0000i	0.574380i
$683$	− 15.0000i	− 0.573959i	−0.957937	−	0.286980i	$-0.907349\pi$
$683$	− 15.0000i	− 0.573959i	0.957937	−	0.286980i	$-0.0926512\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−13.0000	−0.496342
$687$	28.0000i	1.06827i
$688$	− 1.00000i	− 0.0381246i
$689$	12.0000	0.457164
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	− 15.0000i	− 0.570214i
$693$	− 3.00000i	− 0.113961i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	6.00000	0.227429
$697$	− 9.00000i	− 0.340899i
$698$	− 26.0000i	− 0.984115i
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	− 16.0000i	− 0.603881i
$703$	2.00000i	0.0754314i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	21.0000	0.790345
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 6.00000i	− 0.224860i
$713$	30.0000i	1.12351i
$714$	6.00000	0.224544
$715$	0	0
$716$	0	0
$717$	− 18.0000i	− 0.672222i
$718$	36.0000i	1.34351i
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	− 15.0000i	− 0.558242i
$723$	56.0000i	2.08266i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−4.00000	−0.148454
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	− 2.00000i	− 0.0739221i
$733$	− 31.0000i	− 1.14501i	−0.819901	−	0.572506i	$-0.805971\pi$
$733$	− 31.0000i	− 1.14501i	0.819901	−	0.572506i	$-0.194029\pi$
$734$	35.0000	1.29187
$735$	0	0
$736$	−6.00000	−0.221163
$737$	12.0000i	0.442026i
$738$	− 3.00000i	− 0.110432i
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	− 3.00000i	− 0.110133i
$743$	− 51.0000i	− 1.87101i	−0.353315	−	0.935504i	$-0.614946\pi$
$743$	− 51.0000i	− 1.87101i	0.353315	−	0.935504i	$-0.385054\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	22.0000	0.805477
$747$	12.0000i	0.439057i
$748$	9.00000i	0.329073i
$749$	6.00000	0.219235
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	− 12.0000i	− 0.437595i
$753$	− 36.0000i	− 1.31191i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−4.00000	−0.145479
$757$	34.0000i	1.23575i	0.786276	+	0.617876i	$0.212006\pi$
$757$	34.0000i	1.23575i	−0.786276	+	0.617876i	$0.787994\pi$
$758$	16.0000i	0.581146i
$759$	36.0000	1.30672
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	− 40.0000i	− 1.44905i
$763$	− 11.0000i	− 0.398227i
$764$	−3.00000	−0.108536
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	− 14.0000i	− 0.503871i
$773$	− 39.0000i	− 1.40273i	−0.712801	−	0.701366i	$-0.752574\pi$
$773$	− 39.0000i	− 1.40273i	0.712801	−	0.701366i	$-0.247426\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−17.0000	−0.610264
$777$	− 2.00000i	− 0.0717496i
$778$	− 9.00000i	− 0.322666i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	18.0000	0.644091
$782$	18.0000i	0.643679i
$783$	12.0000i	0.428845i
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	− 6.00000i	− 0.213741i
$789$	18.0000	0.640817
$790$	0	0
$791$	−9.00000	−0.320003
$792$	3.00000i	0.106600i
$793$	4.00000i	0.142044i
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−16.0000	−0.567105
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	− 4.00000i	− 0.141598i
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	6.00000i	0.211867i
$803$	− 48.0000i	− 1.69388i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	20.0000	0.704470
$807$	− 36.0000i	− 1.26726i
$808$	6.00000i	0.211079i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	3.00000i	0.105279i
$813$	− 4.00000i	− 0.140286i
$814$	3.00000	0.105150
$815$	0	0
$816$	−6.00000	−0.210042
$817$	2.00000i	0.0699711i
$818$	− 32.0000i	− 1.11885i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	− 24.0000i	− 0.837096i
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	3.00000i	0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$	3.00000i	0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$	6.00000i	0.208514i
$829$	−47.0000	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$830$	0	0
$831$	56.0000	1.94262
$832$	4.00000i	0.138675i
$833$	− 18.0000i	− 0.623663i
$834$	26.0000	0.900306
$835$	0	0
$836$	6.00000	0.207514
$837$	− 20.0000i	− 0.691301i
$838$	36.0000i	1.24360i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 10.0000i	− 0.344623i
$843$	− 48.0000i	− 1.65321i
$844$	−5.00000	−0.172107
$845$	0	0
$846$	−12.0000	−0.412568
$847$	− 2.00000i	− 0.0687208i
$848$	3.00000i	0.103020i
$849$	−8.00000	−0.274559
$850$	0	0
$851$	6.00000	0.205677
$852$	12.0000i	0.411113i
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	−6.00000	−0.205076
$857$	15.0000i	0.512390i	0.966625	+	0.256195i	$0.0824690\pi$
$857$	15.0000i	0.512390i	−0.966625	+	0.256195i	$0.917531\pi$
$858$	− 24.0000i	− 0.819346i
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	15.0000i	0.510902i
$863$	39.0000i	1.32758i	0.747921	+	0.663788i	$0.231052\pi$
$863$	39.0000i	1.32758i	−0.747921	+	0.663788i	$0.768948\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	− 16.0000i	− 0.543388i
$868$	− 5.00000i	− 0.169711i
$869$	−24.0000	−0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	11.0000i	0.372507i
$873$	17.0000i	0.575363i
$874$	12.0000	0.405906
$875$	0	0
$876$	32.0000	1.08118
$877$	13.0000i	0.438979i	0.975615	+	0.219489i	$0.0704391\pi$
$877$	13.0000i	0.438979i	−0.975615	+	0.219489i	$0.929561\pi$
$878$	− 35.0000i	− 1.18119i
$879$	42.0000	1.41662
$880$	0	0
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	− 6.00000i	− 0.202031i
$883$	29.0000i	0.975928i	0.872864	+	0.487964i	$0.162260\pi$
$883$	29.0000i	0.975928i	−0.872864	+	0.487964i	$0.837740\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−30.0000	−1.00787
$887$	− 33.0000i	− 1.10803i	−0.832506	−	0.554016i	$-0.813095\pi$
$887$	− 33.0000i	− 1.10803i	0.832506	−	0.554016i	$-0.186905\pi$
$888$	2.00000i	0.0671156i
$889$	20.0000	0.670778
$890$	0	0
$891$	−33.0000	−1.10554
$892$	− 17.0000i	− 0.569202i
$893$	24.0000i	0.803129i
$894$	−12.0000	−0.401340
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 48.0000i	− 1.60267i
$898$	− 6.00000i	− 0.200223i
$899$	−15.0000	−0.500278
$900$	0	0
$901$	9.00000	0.299833
$902$	9.00000i	0.299667i
$903$	− 2.00000i	− 0.0665558i
$904$	9.00000	0.299336
$905$	0	0
$906$	16.0000	0.531564
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	− 27.0000i	− 0.896026i
$909$	6.00000	0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	4.00000i	0.132453i
$913$	− 36.0000i	− 1.19143i
$914$	17.0000	0.562310
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	− 12.0000i	− 0.396059i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	68.0000	2.24068
$922$	33.0000i	1.08680i
$923$	− 24.0000i	− 0.789970i
$924$	−6.00000	−0.197386
$925$	0	0
$926$	4.00000	0.131448
$927$	4.00000i	0.131377i
$928$	− 3.00000i	− 0.0984798i
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	18.0000i	0.589294i
$934$	27.0000	0.883467
$935$	0	0
$936$	4.00000	0.130744
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	− 4.00000i	− 0.130605i
$939$	52.0000	1.69696
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	26.0000i	0.847126i
$943$	18.0000i	0.586161i
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	− 33.0000i	− 1.07236i	−0.844105	−	0.536178i	$-0.819868\pi$
$947$	− 33.0000i	− 1.07236i	0.844105	−	0.536178i	$-0.180132\pi$
$948$	− 16.0000i	− 0.519656i
$949$	−64.0000	−2.07753
$950$	0	0
$951$	42.0000	1.36194
$952$	− 3.00000i	− 0.0972306i
$953$	− 60.0000i	− 1.94359i	−0.235826	−	0.971795i	$-0.575780\pi$
$953$	− 60.0000i	− 1.94359i	0.235826	−	0.971795i	$-0.424220\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	−9.00000	−0.291081
$957$	18.0000i	0.581857i
$958$	− 24.0000i	− 0.775405i
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	− 4.00000i	− 0.128965i
$963$	6.00000i	0.193347i
$964$	28.0000	0.901819
$965$	0	0
$966$	−12.0000	−0.386094
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	2.00000i	0.0642824i
$969$	12.0000	0.385496
$970$	0	0
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	− 10.0000i	− 0.320750i
$973$	13.0000i	0.416761i
$974$	2.00000	0.0640841
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	45.0000i	1.43968i	0.694141	+	0.719839i	$0.255784\pi$
$977$	45.0000i	1.43968i	−0.694141	+	0.719839i	$0.744216\pi$
$978$	22.0000i	0.703482i
$979$	18.0000	0.575282
$980$	0	0
$981$	11.0000	0.351203
$982$	− 12.0000i	− 0.382935i
$983$	51.0000i	1.62665i	0.581811	+	0.813324i	$0.302344\pi$
$983$	51.0000i	1.62665i	−0.581811	+	0.813324i	$0.697656\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−9.00000	−0.286618
$987$	− 24.0000i	− 0.763928i
$988$	− 8.00000i	− 0.254514i
$989$	6.00000	0.190789
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	5.00000i	0.158750i
$993$	− 52.0000i	− 1.65017i
$994$	−6.00000	−0.190308
$995$	0	0
$996$	24.0000	0.760469
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	− 14.0000i	− 0.443162i
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.g.149.2		2
5.2	odd	4		370.2.a.a.1.1	✓	1
5.3	odd	4		1850.2.a.o.1.1		1
5.4	even	2	inner	1850.2.b.g.149.1		2
15.2	even	4		3330.2.a.v.1.1		1
20.7	even	4		2960.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.a.1.1	✓	1	5.2	odd	4
1850.2.a.o.1.1		1	5.3	odd	4
1850.2.b.g.149.1		2	5.4	even	2	inner
1850.2.b.g.149.2		2	1.1	even	1	trivial
2960.2.a.j.1.1		1	20.7	even	4
3330.2.a.v.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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +3.00000i q^{22} +6.00000i q^{23} -2.00000 q^{24} +4.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -2.00000i q^{38} -8.00000 q^{39} +3.00000 q^{41} +2.00000i q^{42} -1.00000i q^{43} -3.00000 q^{44} -6.00000 q^{46} -12.0000i q^{47} -2.00000i q^{48} +6.00000 q^{49} -6.00000 q^{51} +4.00000i q^{52} +3.00000i q^{53} +4.00000 q^{54} +1.00000 q^{56} +4.00000i q^{57} -3.00000i q^{58} -1.00000 q^{61} +5.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +4.00000i q^{67} +3.00000i q^{68} +12.0000 q^{69} +6.00000 q^{71} +1.00000i q^{72} -16.0000i q^{73} +1.00000 q^{74} +2.00000 q^{76} +3.00000i q^{77} -8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} +3.00000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +1.00000 q^{86} +6.00000i q^{87} -3.00000i q^{88} +6.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} -10.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} -17.0000i q^{97} +6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 6 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} - 6 q^{29} + 10 q^{31} + 6 q^{34} + 2 q^{36} - 16 q^{39} + 6 q^{41} - 6 q^{44} - 12 q^{46} + 12 q^{49} - 12 q^{51} + 8 q^{54} + 2 q^{56} - 2 q^{61} - 2 q^{64} + 12 q^{66} + 24 q^{69} + 12 q^{71} + 2 q^{74} + 4 q^{76} - 16 q^{79} - 22 q^{81} - 4 q^{84} + 2 q^{86} + 12 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 6 * q^11 - 2 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^21 - 4 * q^24 + 8 * q^26 - 6 * q^29 + 10 * q^31 + 6 * q^34 + 2 * q^36 - 16 * q^39 + 6 * q^41 - 6 * q^44 - 12 * q^46 + 12 * q^49 - 12 * q^51 + 8 * q^54 + 2 * q^56 - 2 * q^61 - 2 * q^64 + 12 * q^66 + 24 * q^69 + 12 * q^71 + 2 * q^74 + 4 * q^76 - 16 * q^79 - 22 * q^81 - 4 * q^84 + 2 * q^86 + 12 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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