LMFDB - Embedded newform 2850.2.d.u.799.4

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.u.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.73205i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.73205i q^{7} -1.00000i q^{8} -1.00000 q^{9} +5.19615 q^{11} -1.00000i q^{12} -4.73205i q^{13} -2.73205 q^{14} +1.00000 q^{16} -2.73205i q^{17} -1.00000i q^{18} -1.00000 q^{19} -2.73205 q^{21} +5.19615i q^{22} -8.46410i q^{23} +1.00000 q^{24} +4.73205 q^{26} -1.00000i q^{27} -2.73205i q^{28} +9.19615 q^{29} -5.92820 q^{31} +1.00000i q^{32} +5.19615i q^{33} +2.73205 q^{34} +1.00000 q^{36} -8.92820i q^{37} -1.00000i q^{38} +4.73205 q^{39} -2.73205i q^{42} -8.73205i q^{43} -5.19615 q^{44} +8.46410 q^{46} -3.46410i q^{47} +1.00000i q^{48} -0.464102 q^{49} +2.73205 q^{51} +4.73205i q^{52} +4.66025i q^{53} +1.00000 q^{54} +2.73205 q^{56} -1.00000i q^{57} +9.19615i q^{58} -2.19615 q^{59} +8.26795 q^{61} -5.92820i q^{62} -2.73205i q^{63} -1.00000 q^{64} -5.19615 q^{66} -5.19615i q^{67} +2.73205i q^{68} +8.46410 q^{69} -10.1962 q^{71} +1.00000i q^{72} -14.4641i q^{73} +8.92820 q^{74} +1.00000 q^{76} +14.1962i q^{77} +4.73205i q^{78} -5.92820 q^{79} +1.00000 q^{81} -16.6603i q^{83} +2.73205 q^{84} +8.73205 q^{86} +9.19615i q^{87} -5.19615i q^{88} -3.92820 q^{89} +12.9282 q^{91} +8.46410i q^{92} -5.92820i q^{93} +3.46410 q^{94} -1.00000 q^{96} +17.1244i q^{97} -0.464102i q^{98} -5.19615 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 4 q^{14} + 4 q^{16} - 4 q^{19} - 4 q^{21} + 4 q^{24} + 12 q^{26} + 16 q^{29} + 4 q^{31} + 4 q^{34} + 4 q^{36} + 12 q^{39} + 20 q^{46} + 12 q^{49} + 4 q^{51} + 4 q^{54} + 4 q^{56} + 12 q^{59} + 40 q^{61} - 4 q^{64} + 20 q^{69} - 20 q^{71} + 8 q^{74} + 4 q^{76} + 4 q^{79} + 4 q^{81} + 4 q^{84} + 28 q^{86} + 12 q^{89} + 24 q^{91} - 4 q^{96}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 4 * q^14 + 4 * q^16 - 4 * q^19 - 4 * q^21 + 4 * q^24 + 12 * q^26 + 16 * q^29 + 4 * q^31 + 4 * q^34 + 4 * q^36 + 12 * q^39 + 20 * q^46 + 12 * q^49 + 4 * q^51 + 4 * q^54 + 4 * q^56 + 12 * q^59 + 40 * q^61 - 4 * q^64 + 20 * q^69 - 20 * q^71 + 8 * q^74 + 4 * q^76 + 4 * q^79 + 4 * q^81 + 4 * q^84 + 28 * q^86 + 12 * q^89 + 24 * q^91 - 4 * 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
$7$	2.73205i	1.03262i	0.856402	+	0.516309i	$0.172694\pi$
$7$	2.73205i	1.03262i	−0.856402	+	0.516309i	$0.827306\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.19615	1.56670	0.783349	−	0.621582i	$-0.213510\pi$
$11$	5.19615	1.56670	0.783349	+	0.621582i	$0.213510\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 4.73205i	− 1.31243i	−0.754572	−	0.656217i	$-0.772155\pi$
$13$	− 4.73205i	− 1.31243i	0.754572	−	0.656217i	$-0.227845\pi$
$14$	−2.73205	−0.730171
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.73205i	− 0.662620i	−0.943522	−	0.331310i	$-0.892509\pi$
$17$	− 2.73205i	− 0.662620i	0.943522	−	0.331310i	$-0.107491\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.73205	−0.596182
$22$	5.19615i	1.10782i
$23$	− 8.46410i	− 1.76489i	−0.470418	−	0.882444i	$-0.655897\pi$
$23$	− 8.46410i	− 1.76489i	0.470418	−	0.882444i	$-0.344103\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.73205	0.928032
$27$	− 1.00000i	− 0.192450i
$28$	− 2.73205i	− 0.516309i
$29$	9.19615	1.70768	0.853841	−	0.520533i	$-0.174267\pi$
$29$	9.19615	1.70768	0.853841	+	0.520533i	$0.174267\pi$
$30$	0	0
$31$	−5.92820	−1.06474	−0.532368	−	0.846513i	$-0.678698\pi$
$31$	−5.92820	−1.06474	−0.532368	+	0.846513i	$0.678698\pi$
$32$	1.00000i	0.176777i
$33$	5.19615i	0.904534i
$34$	2.73205	0.468543
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.92820i	− 1.46779i	−0.679264	−	0.733894i	$-0.737701\pi$
$37$	− 8.92820i	− 1.46779i	0.679264	−	0.733894i	$-0.262299\pi$
$38$	− 1.00000i	− 0.162221i
$39$	4.73205	0.757735
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	− 2.73205i	− 0.421565i
$43$	− 8.73205i	− 1.33163i	−0.746119	−	0.665813i	$-0.768085\pi$
$43$	− 8.73205i	− 1.33163i	0.746119	−	0.665813i	$-0.231915\pi$
$44$	−5.19615	−0.783349
$45$	0	0
$46$	8.46410	1.24796
$47$	− 3.46410i	− 0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$	− 3.46410i	− 0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	1.00000i	0.144338i
$49$	−0.464102	−0.0663002
$50$	0	0
$51$	2.73205	0.382564
$52$	4.73205i	0.656217i
$53$	4.66025i	0.640135i	0.947395	+	0.320068i	$0.103706\pi$
$53$	4.66025i	0.640135i	−0.947395	+	0.320068i	$0.896294\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.73205	0.365086
$57$	− 1.00000i	− 0.132453i
$58$	9.19615i	1.20751i
$59$	−2.19615	−0.285915	−0.142957	−	0.989729i	$-0.545661\pi$
$59$	−2.19615	−0.285915	−0.142957	+	0.989729i	$0.545661\pi$
$60$	0	0
$61$	8.26795	1.05860	0.529301	−	0.848434i	$-0.322454\pi$
$61$	8.26795	1.05860	0.529301	+	0.848434i	$0.322454\pi$
$62$	− 5.92820i	− 0.752883i
$63$	− 2.73205i	− 0.344206i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−5.19615	−0.639602
$67$	− 5.19615i	− 0.634811i	−0.948290	−	0.317406i	$-0.897188\pi$
$67$	− 5.19615i	− 0.634811i	0.948290	−	0.317406i	$-0.102812\pi$
$68$	2.73205i	0.331310i
$69$	8.46410	1.01896
$70$	0	0
$71$	−10.1962	−1.21006	−0.605030	−	0.796202i	$-0.706839\pi$
$71$	−10.1962	−1.21006	−0.605030	+	0.796202i	$0.706839\pi$
$72$	1.00000i	0.117851i
$73$	− 14.4641i	− 1.69290i	−0.532472	−	0.846448i	$-0.678737\pi$
$73$	− 14.4641i	− 1.69290i	0.532472	−	0.846448i	$-0.321263\pi$
$74$	8.92820	1.03788
$75$	0	0
$76$	1.00000	0.114708
$77$	14.1962i	1.61780i
$78$	4.73205i	0.535799i
$79$	−5.92820	−0.666975	−0.333487	−	0.942755i	$-0.608225\pi$
$79$	−5.92820	−0.666975	−0.333487	+	0.942755i	$0.608225\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 16.6603i	− 1.82870i	−0.404924	−	0.914350i	$-0.632702\pi$
$83$	− 16.6603i	− 1.82870i	0.404924	−	0.914350i	$-0.367298\pi$
$84$	2.73205	0.298091
$85$	0	0
$86$	8.73205	0.941601
$87$	9.19615i	0.985931i
$88$	− 5.19615i	− 0.553912i
$89$	−3.92820	−0.416389	−0.208194	−	0.978087i	$-0.566759\pi$
$89$	−3.92820	−0.416389	−0.208194	+	0.978087i	$0.566759\pi$
$90$	0	0
$91$	12.9282	1.35524
$92$	8.46410i	0.882444i
$93$	− 5.92820i	− 0.614726i
$94$	3.46410	0.357295
$95$	0	0
$96$	−1.00000	−0.102062
$97$	17.1244i	1.73871i	0.494184	+	0.869357i	$0.335467\pi$
$97$	17.1244i	1.73871i	−0.494184	+	0.869357i	$0.664533\pi$
$98$	− 0.464102i	− 0.0468813i
$99$	−5.19615	−0.522233
$100$	0	0
$101$	−11.6603	−1.16024	−0.580119	−	0.814532i	$-0.696994\pi$
$101$	−11.6603	−1.16024	−0.580119	+	0.814532i	$0.696994\pi$
$102$	2.73205i	0.270513i
$103$	14.8564i	1.46385i	0.681388	+	0.731923i	$0.261377\pi$
$103$	14.8564i	1.46385i	−0.681388	+	0.731923i	$0.738623\pi$
$104$	−4.73205	−0.464016
$105$	0	0
$106$	−4.66025	−0.452644
$107$	4.19615i	0.405657i	0.979214	+	0.202829i	$0.0650135\pi$
$107$	4.19615i	0.405657i	−0.979214	+	0.202829i	$0.934987\pi$
$108$	1.00000i	0.0962250i
$109$	10.3923	0.995402	0.497701	−	0.867349i	$-0.334178\pi$
$109$	10.3923	0.995402	0.497701	+	0.867349i	$0.334178\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	2.73205i	0.258155i
$113$	14.3205i	1.34716i	0.739114	+	0.673580i	$0.235244\pi$
$113$	14.3205i	1.34716i	−0.739114	+	0.673580i	$0.764756\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−9.19615	−0.853841
$117$	4.73205i	0.437478i
$118$	− 2.19615i	− 0.202172i
$119$	7.46410	0.684233
$120$	0	0
$121$	16.0000	1.45455
$122$	8.26795i	0.748545i
$123$	0	0
$124$	5.92820	0.532368
$125$	0	0
$126$	2.73205	0.243390
$127$	− 9.39230i	− 0.833432i	−0.909037	−	0.416716i	$-0.863181\pi$
$127$	− 9.39230i	− 0.833432i	0.909037	−	0.416716i	$-0.136819\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	8.73205	0.768814
$130$	0	0
$131$	−11.1962	−0.978212	−0.489106	−	0.872224i	$-0.662677\pi$
$131$	−11.1962	−0.978212	−0.489106	+	0.872224i	$0.662677\pi$
$132$	− 5.19615i	− 0.452267i
$133$	− 2.73205i	− 0.236899i
$134$	5.19615	0.448879
$135$	0	0
$136$	−2.73205	−0.234271
$137$	16.3923i	1.40049i	0.713903	+	0.700245i	$0.246926\pi$
$137$	16.3923i	1.40049i	−0.713903	+	0.700245i	$0.753074\pi$
$138$	8.46410i	0.720512i
$139$	3.80385	0.322638	0.161319	−	0.986902i	$-0.448425\pi$
$139$	3.80385	0.322638	0.161319	+	0.986902i	$0.448425\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	− 10.1962i	− 0.855642i
$143$	− 24.5885i	− 2.05619i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.4641	1.19706
$147$	− 0.464102i	− 0.0382785i
$148$	8.92820i	0.733894i
$149$	21.8564	1.79055	0.895273	−	0.445517i	$-0.146980\pi$
$149$	21.8564	1.79055	0.895273	+	0.445517i	$0.146980\pi$
$150$	0	0
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	1.00000i	0.0811107i
$153$	2.73205i	0.220873i
$154$	−14.1962	−1.14396
$155$	0	0
$156$	−4.73205	−0.378867
$157$	21.8564i	1.74433i	0.489211	+	0.872166i	$0.337285\pi$
$157$	21.8564i	1.74433i	−0.489211	+	0.872166i	$0.662715\pi$
$158$	− 5.92820i	− 0.471623i
$159$	−4.66025	−0.369582
$160$	0	0
$161$	23.1244	1.82245
$162$	1.00000i	0.0785674i
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	0	0
$166$	16.6603	1.29309
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	2.73205i	0.210782i
$169$	−9.39230	−0.722485
$170$	0	0
$171$	1.00000	0.0764719
$172$	8.73205i	0.665813i
$173$	− 22.6603i	− 1.72283i	−0.507904	−	0.861414i	$-0.669580\pi$
$173$	− 22.6603i	− 1.72283i	0.507904	−	0.861414i	$-0.330420\pi$
$174$	−9.19615	−0.697159
$175$	0	0
$176$	5.19615	0.391675
$177$	− 2.19615i	− 0.165073i
$178$	− 3.92820i	− 0.294431i
$179$	9.85641	0.736702	0.368351	−	0.929687i	$-0.379922\pi$
$179$	9.85641	0.736702	0.368351	+	0.929687i	$0.379922\pi$
$180$	0	0
$181$	19.2679	1.43218	0.716088	−	0.698010i	$-0.245931\pi$
$181$	19.2679	1.43218	0.716088	+	0.698010i	$0.245931\pi$
$182$	12.9282i	0.958302i
$183$	8.26795i	0.611184i
$184$	−8.46410	−0.623982
$185$	0	0
$186$	5.92820	0.434677
$187$	− 14.1962i	− 1.03813i
$188$	3.46410i	0.252646i
$189$	2.73205	0.198727
$190$	0	0
$191$	−24.4641	−1.77016	−0.885080	−	0.465439i	$-0.845897\pi$
$191$	−24.4641	−1.77016	−0.885080	+	0.465439i	$0.845897\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	10.1962i	0.733935i	0.930234	+	0.366968i	$0.119604\pi$
$193$	10.1962i	0.733935i	−0.930234	+	0.366968i	$0.880396\pi$
$194$	−17.1244	−1.22946
$195$	0	0
$196$	0.464102	0.0331501
$197$	0.339746i	0.0242059i	0.999927	+	0.0121029i	$0.00385258\pi$
$197$	0.339746i	0.0242059i	−0.999927	+	0.0121029i	$0.996147\pi$
$198$	− 5.19615i	− 0.369274i
$199$	−3.07180	−0.217754	−0.108877	−	0.994055i	$-0.534725\pi$
$199$	−3.07180	−0.217754	−0.108877	+	0.994055i	$0.534725\pi$
$200$	0	0
$201$	5.19615	0.366508
$202$	− 11.6603i	− 0.820413i
$203$	25.1244i	1.76338i
$204$	−2.73205	−0.191282
$205$	0	0
$206$	−14.8564	−1.03509
$207$	8.46410i	0.588296i
$208$	− 4.73205i	− 0.328109i
$209$	−5.19615	−0.359425
$210$	0	0
$211$	7.19615	0.495404	0.247702	−	0.968836i	$-0.420325\pi$
$211$	7.19615	0.495404	0.247702	+	0.968836i	$0.420325\pi$
$212$	− 4.66025i	− 0.320068i
$213$	− 10.1962i	− 0.698629i
$214$	−4.19615	−0.286843
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 16.1962i	− 1.09947i
$218$	10.3923i	0.703856i
$219$	14.4641	0.977393
$220$	0	0
$221$	−12.9282	−0.869645
$222$	8.92820i	0.599222i
$223$	− 17.9282i	− 1.20056i	−0.799789	−	0.600281i	$-0.795056\pi$
$223$	− 17.9282i	− 1.20056i	0.799789	−	0.600281i	$-0.204944\pi$
$224$	−2.73205	−0.182543
$225$	0	0
$226$	−14.3205	−0.952586
$227$	2.00000i	0.132745i	0.997795	+	0.0663723i	$0.0211425\pi$
$227$	2.00000i	0.132745i	−0.997795	+	0.0663723i	$0.978857\pi$
$228$	1.00000i	0.0662266i
$229$	−5.19615	−0.343371	−0.171686	−	0.985152i	$-0.554921\pi$
$229$	−5.19615	−0.343371	−0.171686	+	0.985152i	$0.554921\pi$
$230$	0	0
$231$	−14.1962	−0.934038
$232$	− 9.19615i	− 0.603757i
$233$	17.6603i	1.15696i	0.815696	+	0.578481i	$0.196354\pi$
$233$	17.6603i	1.15696i	−0.815696	+	0.578481i	$0.803646\pi$
$234$	−4.73205	−0.309344
$235$	0	0
$236$	2.19615	0.142957
$237$	− 5.92820i	− 0.385078i
$238$	7.46410i	0.483826i
$239$	7.07180	0.457437	0.228718	−	0.973493i	$-0.426547\pi$
$239$	7.07180	0.457437	0.228718	+	0.973493i	$0.426547\pi$
$240$	0	0
$241$	7.80385	0.502690	0.251345	−	0.967898i	$-0.419127\pi$
$241$	7.80385	0.502690	0.251345	+	0.967898i	$0.419127\pi$
$242$	16.0000i	1.02852i
$243$	1.00000i	0.0641500i
$244$	−8.26795	−0.529301
$245$	0	0
$246$	0	0
$247$	4.73205i	0.301093i
$248$	5.92820i	0.376441i
$249$	16.6603	1.05580
$250$	0	0
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	2.73205i	0.172103i
$253$	− 43.9808i	− 2.76505i
$254$	9.39230	0.589326
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 15.9282i	− 0.993574i	−0.867872	−	0.496787i	$-0.834513\pi$
$257$	− 15.9282i	− 0.993574i	0.867872	−	0.496787i	$-0.165487\pi$
$258$	8.73205i	0.543634i
$259$	24.3923	1.51566
$260$	0	0
$261$	−9.19615	−0.569228
$262$	− 11.1962i	− 0.691701i
$263$	− 12.4641i	− 0.768569i	−0.923215	−	0.384285i	$-0.874448\pi$
$263$	− 12.4641i	− 0.768569i	0.923215	−	0.384285i	$-0.125552\pi$
$264$	5.19615	0.319801
$265$	0	0
$266$	2.73205	0.167513
$267$	− 3.92820i	− 0.240402i
$268$	5.19615i	0.317406i
$269$	−15.4641	−0.942863	−0.471431	−	0.881903i	$-0.656263\pi$
$269$	−15.4641	−0.942863	−0.471431	+	0.881903i	$0.656263\pi$
$270$	0	0
$271$	−25.1244	−1.52620	−0.763098	−	0.646283i	$-0.776323\pi$
$271$	−25.1244	−1.52620	−0.763098	+	0.646283i	$0.776323\pi$
$272$	− 2.73205i	− 0.165655i
$273$	12.9282i	0.782450i
$274$	−16.3923	−0.990295
$275$	0	0
$276$	−8.46410	−0.509479
$277$	− 24.5167i	− 1.47306i	−0.676403	−	0.736532i	$-0.736462\pi$
$277$	− 24.5167i	− 1.47306i	0.676403	−	0.736532i	$-0.263538\pi$
$278$	3.80385i	0.228140i
$279$	5.92820	0.354912
$280$	0	0
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	3.46410i	0.206284i
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	10.1962	0.605030
$285$	0	0
$286$	24.5885	1.45395
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	9.53590	0.560935
$290$	0	0
$291$	−17.1244	−1.00385
$292$	14.4641i	0.846448i
$293$	26.6603i	1.55751i	0.627329	+	0.778754i	$0.284148\pi$
$293$	26.6603i	1.55751i	−0.627329	+	0.778754i	$0.715852\pi$
$294$	0.464102	0.0270670
$295$	0	0
$296$	−8.92820	−0.518941
$297$	− 5.19615i	− 0.301511i
$298$	21.8564i	1.26611i
$299$	−40.0526	−2.31630
$300$	0	0
$301$	23.8564	1.37506
$302$	10.3923i	0.598010i
$303$	− 11.6603i	− 0.669864i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−2.73205	−0.156181
$307$	4.66025i	0.265975i	0.991118	+	0.132987i	$0.0424570\pi$
$307$	4.66025i	0.265975i	−0.991118	+	0.132987i	$0.957543\pi$
$308$	− 14.1962i	− 0.808901i
$309$	−14.8564	−0.845151
$310$	0	0
$311$	−15.4641	−0.876889	−0.438444	−	0.898758i	$-0.644470\pi$
$311$	−15.4641	−0.876889	−0.438444	+	0.898758i	$0.644470\pi$
$312$	− 4.73205i	− 0.267900i
$313$	4.85641i	0.274500i	0.990536	+	0.137250i	$0.0438264\pi$
$313$	4.85641i	0.274500i	−0.990536	+	0.137250i	$0.956174\pi$
$314$	−21.8564	−1.23343
$315$	0	0
$316$	5.92820	0.333487
$317$	23.0526i	1.29476i	0.762167	+	0.647380i	$0.224135\pi$
$317$	23.0526i	1.29476i	−0.762167	+	0.647380i	$0.775865\pi$
$318$	− 4.66025i	− 0.261334i
$319$	47.7846	2.67542
$320$	0	0
$321$	−4.19615	−0.234206
$322$	23.1244i	1.28867i
$323$	2.73205i	0.152015i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−10.0000	−0.553849
$327$	10.3923i	0.574696i
$328$	0	0
$329$	9.46410	0.521773
$330$	0	0
$331$	19.1962	1.05512	0.527558	−	0.849519i	$-0.323108\pi$
$331$	19.1962	1.05512	0.527558	+	0.849519i	$0.323108\pi$
$332$	16.6603i	0.914350i
$333$	8.92820i	0.489263i
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−2.73205	−0.149046
$337$	15.3205i	0.834561i	0.908778	+	0.417281i	$0.137017\pi$
$337$	15.3205i	0.834561i	−0.908778	+	0.417281i	$0.862983\pi$
$338$	− 9.39230i	− 0.510874i
$339$	−14.3205	−0.777783
$340$	0	0
$341$	−30.8038	−1.66812
$342$	1.00000i	0.0540738i
$343$	17.8564i	0.964155i
$344$	−8.73205	−0.470801
$345$	0	0
$346$	22.6603	1.21822
$347$	1.07180i	0.0575371i	0.999586	+	0.0287685i	$0.00915857\pi$
$347$	1.07180i	0.0575371i	−0.999586	+	0.0287685i	$0.990841\pi$
$348$	− 9.19615i	− 0.492966i
$349$	13.5885	0.727373	0.363687	−	0.931521i	$-0.381518\pi$
$349$	13.5885	0.727373	0.363687	+	0.931521i	$0.381518\pi$
$350$	0	0
$351$	−4.73205	−0.252578
$352$	5.19615i	0.276956i
$353$	3.41154i	0.181578i	0.995870	+	0.0907890i	$0.0289389\pi$
$353$	3.41154i	0.181578i	−0.995870	+	0.0907890i	$0.971061\pi$
$354$	2.19615	0.116724
$355$	0	0
$356$	3.92820	0.208194
$357$	7.46410i	0.395042i
$358$	9.85641i	0.520927i
$359$	−19.8564	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$359$	−19.8564	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	19.2679i	1.01270i
$363$	16.0000i	0.839782i
$364$	−12.9282	−0.677622
$365$	0	0
$366$	−8.26795	−0.432173
$367$	29.5167i	1.54076i	0.637587	+	0.770379i	$0.279933\pi$
$367$	29.5167i	1.54076i	−0.637587	+	0.770379i	$0.720067\pi$
$368$	− 8.46410i	− 0.441222i
$369$	0	0
$370$	0	0
$371$	−12.7321	−0.661015
$372$	5.92820i	0.307363i
$373$	− 27.6603i	− 1.43219i	−0.698001	−	0.716097i	$-0.745927\pi$
$373$	− 27.6603i	− 1.43219i	0.698001	−	0.716097i	$-0.254073\pi$
$374$	14.1962	0.734066
$375$	0	0
$376$	−3.46410	−0.178647
$377$	− 43.5167i	− 2.24122i
$378$	2.73205i	0.140522i
$379$	29.1769	1.49872	0.749359	−	0.662164i	$-0.230362\pi$
$379$	29.1769	1.49872	0.749359	+	0.662164i	$0.230362\pi$
$380$	0	0
$381$	9.39230	0.481182
$382$	− 24.4641i	− 1.25169i
$383$	− 4.05256i	− 0.207076i	−0.994625	−	0.103538i	$-0.966984\pi$
$383$	− 4.05256i	− 0.207076i	0.994625	−	0.103538i	$-0.0330164\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−10.1962	−0.518970
$387$	8.73205i	0.443875i
$388$	− 17.1244i	− 0.869357i
$389$	−12.9282	−0.655486	−0.327743	−	0.944767i	$-0.606288\pi$
$389$	−12.9282	−0.655486	−0.327743	+	0.944767i	$0.606288\pi$
$390$	0	0
$391$	−23.1244	−1.16945
$392$	0.464102i	0.0234407i
$393$	− 11.1962i	− 0.564771i
$394$	−0.339746	−0.0171162
$395$	0	0
$396$	5.19615	0.261116
$397$	− 34.1244i	− 1.71265i	−0.516435	−	0.856326i	$-0.672741\pi$
$397$	− 34.1244i	− 1.71265i	0.516435	−	0.856326i	$-0.327259\pi$
$398$	− 3.07180i	− 0.153975i
$399$	2.73205	0.136774
$400$	0	0
$401$	18.7128	0.934473	0.467237	−	0.884132i	$-0.345250\pi$
$401$	18.7128	0.934473	0.467237	+	0.884132i	$0.345250\pi$
$402$	5.19615i	0.259161i
$403$	28.0526i	1.39740i
$404$	11.6603	0.580119
$405$	0	0
$406$	−25.1244	−1.24690
$407$	− 46.3923i	− 2.29958i
$408$	− 2.73205i	− 0.135257i
$409$	−17.3205	−0.856444	−0.428222	−	0.903674i	$-0.640860\pi$
$409$	−17.3205	−0.856444	−0.428222	+	0.903674i	$0.640860\pi$
$410$	0	0
$411$	−16.3923	−0.808573
$412$	− 14.8564i	− 0.731923i
$413$	− 6.00000i	− 0.295241i
$414$	−8.46410	−0.415988
$415$	0	0
$416$	4.73205	0.232008
$417$	3.80385i	0.186275i
$418$	− 5.19615i	− 0.254152i
$419$	5.85641	0.286104	0.143052	−	0.989715i	$-0.454308\pi$
$419$	5.85641	0.286104	0.143052	+	0.989715i	$0.454308\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	7.19615i	0.350303i
$423$	3.46410i	0.168430i
$424$	4.66025	0.226322
$425$	0	0
$426$	10.1962	0.494005
$427$	22.5885i	1.09313i
$428$	− 4.19615i	− 0.202829i
$429$	24.5885	1.18714
$430$	0	0
$431$	−1.60770	−0.0774400	−0.0387200	−	0.999250i	$-0.512328\pi$
$431$	−1.60770	−0.0774400	−0.0387200	+	0.999250i	$0.512328\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 5.80385i	− 0.278915i	−0.990228	−	0.139458i	$-0.955464\pi$
$433$	− 5.80385i	− 0.278915i	0.990228	−	0.139458i	$-0.0445359\pi$
$434$	16.1962	0.777440
$435$	0	0
$436$	−10.3923	−0.497701
$437$	8.46410i	0.404893i
$438$	14.4641i	0.691122i
$439$	−3.39230	−0.161906	−0.0809529	−	0.996718i	$-0.525796\pi$
$439$	−3.39230	−0.161906	−0.0809529	+	0.996718i	$0.525796\pi$
$440$	0	0
$441$	0.464102	0.0221001
$442$	− 12.9282i	− 0.614932i
$443$	20.5167i	0.974776i	0.873186	+	0.487388i	$0.162050\pi$
$443$	20.5167i	0.974776i	−0.873186	+	0.487388i	$0.837950\pi$
$444$	−8.92820	−0.423714
$445$	0	0
$446$	17.9282	0.848925
$447$	21.8564i	1.03377i
$448$	− 2.73205i	− 0.129077i
$449$	−0.464102	−0.0219023	−0.0109512	−	0.999940i	$-0.503486\pi$
$449$	−0.464102	−0.0219023	−0.0109512	+	0.999940i	$0.503486\pi$
$450$	0	0
$451$	0	0
$452$	− 14.3205i	− 0.673580i
$453$	10.3923i	0.488273i
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	− 32.0000i	− 1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$	− 32.0000i	− 1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$	− 5.19615i	− 0.242800i
$459$	−2.73205	−0.127521
$460$	0	0
$461$	−1.85641	−0.0864615	−0.0432307	−	0.999065i	$-0.513765\pi$
$461$	−1.85641	−0.0864615	−0.0432307	+	0.999065i	$0.513765\pi$
$462$	− 14.1962i	− 0.660465i
$463$	− 11.6077i	− 0.539455i	−0.962937	−	0.269728i	$-0.913066\pi$
$463$	− 11.6077i	− 0.539455i	0.962937	−	0.269728i	$-0.0869337\pi$
$464$	9.19615	0.426921
$465$	0	0
$466$	−17.6603	−0.818095
$467$	16.6603i	0.770945i	0.922719	+	0.385472i	$0.125961\pi$
$467$	16.6603i	0.770945i	−0.922719	+	0.385472i	$0.874039\pi$
$468$	− 4.73205i	− 0.218739i
$469$	14.1962	0.655517
$470$	0	0
$471$	−21.8564	−1.00709
$472$	2.19615i	0.101086i
$473$	− 45.3731i	− 2.08626i
$474$	5.92820	0.272291
$475$	0	0
$476$	−7.46410	−0.342117
$477$	− 4.66025i	− 0.213378i
$478$	7.07180i	0.323456i
$479$	16.6077	0.758825	0.379412	−	0.925228i	$-0.376126\pi$
$479$	16.6077	0.758825	0.379412	+	0.925228i	$0.376126\pi$
$480$	0	0
$481$	−42.2487	−1.92638
$482$	7.80385i	0.355456i
$483$	23.1244i	1.05219i
$484$	−16.0000	−0.727273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 13.0718i	− 0.592340i	−0.955135	−	0.296170i	$-0.904291\pi$
$487$	− 13.0718i	− 0.592340i	0.955135	−	0.296170i	$-0.0957094\pi$
$488$	− 8.26795i	− 0.374272i
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−34.9282	−1.57629	−0.788144	−	0.615491i	$-0.788958\pi$
$491$	−34.9282	−1.57629	−0.788144	+	0.615491i	$0.788958\pi$
$492$	0	0
$493$	− 25.1244i	− 1.13154i
$494$	−4.73205	−0.212905
$495$	0	0
$496$	−5.92820	−0.266184
$497$	− 27.8564i	− 1.24953i
$498$	16.6603i	0.746564i
$499$	−13.1244	−0.587527	−0.293763	−	0.955878i	$-0.594908\pi$
$499$	−13.1244	−0.587527	−0.293763	+	0.955878i	$0.594908\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	3.46410i	0.154610i
$503$	12.7846i	0.570038i	0.958522	+	0.285019i	$0.0919998\pi$
$503$	12.7846i	0.570038i	−0.958522	+	0.285019i	$0.908000\pi$
$504$	−2.73205	−0.121695
$505$	0	0
$506$	43.9808	1.95518
$507$	− 9.39230i	− 0.417127i
$508$	9.39230i	0.416716i
$509$	−0.267949	−0.0118766	−0.00593832	−	0.999982i	$-0.501890\pi$
$509$	−0.267949	−0.0118766	−0.00593832	+	0.999982i	$0.501890\pi$
$510$	0	0
$511$	39.5167	1.74811
$512$	1.00000i	0.0441942i
$513$	1.00000i	0.0441511i
$514$	15.9282	0.702563
$515$	0	0
$516$	−8.73205	−0.384407
$517$	− 18.0000i	− 0.791639i
$518$	24.3923i	1.07174i
$519$	22.6603	0.994675
$520$	0	0
$521$	3.67949	0.161201	0.0806007	−	0.996746i	$-0.474316\pi$
$521$	3.67949	0.161201	0.0806007	+	0.996746i	$0.474316\pi$
$522$	− 9.19615i	− 0.402505i
$523$	− 17.8564i	− 0.780806i	−0.920644	−	0.390403i	$-0.872336\pi$
$523$	− 17.8564i	− 0.780806i	0.920644	−	0.390403i	$-0.127664\pi$
$524$	11.1962	0.489106
$525$	0	0
$526$	12.4641	0.543461
$527$	16.1962i	0.705515i
$528$	5.19615i	0.226134i
$529$	−48.6410	−2.11483
$530$	0	0
$531$	2.19615	0.0953049
$532$	2.73205i	0.118449i
$533$	0	0
$534$	3.92820	0.169990
$535$	0	0
$536$	−5.19615	−0.224440
$537$	9.85641i	0.425335i
$538$	− 15.4641i	− 0.666705i
$539$	−2.41154	−0.103872
$540$	0	0
$541$	8.66025	0.372333	0.186167	−	0.982518i	$-0.440394\pi$
$541$	8.66025	0.372333	0.186167	+	0.982518i	$0.440394\pi$
$542$	− 25.1244i	− 1.07918i
$543$	19.2679i	0.826867i
$544$	2.73205	0.117136
$545$	0	0
$546$	−12.9282	−0.553276
$547$	3.58846i	0.153431i	0.997053	+	0.0767157i	$0.0244434\pi$
$547$	3.58846i	0.153431i	−0.997053	+	0.0767157i	$0.975557\pi$
$548$	− 16.3923i	− 0.700245i
$549$	−8.26795	−0.352867
$550$	0	0
$551$	−9.19615	−0.391769
$552$	− 8.46410i	− 0.360256i
$553$	− 16.1962i	− 0.688730i
$554$	24.5167	1.04161
$555$	0	0
$556$	−3.80385	−0.161319
$557$	− 38.5885i	− 1.63505i	−0.575896	−	0.817523i	$-0.695347\pi$
$557$	− 38.5885i	− 1.63505i	0.575896	−	0.817523i	$-0.304653\pi$
$558$	5.92820i	0.250961i
$559$	−41.3205	−1.74767
$560$	0	0
$561$	14.1962	0.599362
$562$	− 11.0000i	− 0.464007i
$563$	32.1962i	1.35691i	0.734644	+	0.678453i	$0.237349\pi$
$563$	32.1962i	1.35691i	−0.734644	+	0.678453i	$0.762651\pi$
$564$	−3.46410	−0.145865
$565$	0	0
$566$	−4.00000	−0.168133
$567$	2.73205i	0.114735i
$568$	10.1962i	0.427821i
$569$	−13.7128	−0.574871	−0.287436	−	0.957800i	$-0.592803\pi$
$569$	−13.7128	−0.574871	−0.287436	+	0.957800i	$0.592803\pi$
$570$	0	0
$571$	−10.7321	−0.449122	−0.224561	−	0.974460i	$-0.572095\pi$
$571$	−10.7321	−0.449122	−0.224561	+	0.974460i	$0.572095\pi$
$572$	24.5885i	1.02810i
$573$	− 24.4641i	− 1.02200i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 20.3205i	− 0.845954i	−0.906140	−	0.422977i	$-0.860985\pi$
$577$	− 20.3205i	− 0.845954i	0.906140	−	0.422977i	$-0.139015\pi$
$578$	9.53590i	0.396641i
$579$	−10.1962	−0.423738
$580$	0	0
$581$	45.5167	1.88835
$582$	− 17.1244i	− 0.709827i
$583$	24.2154i	1.00290i
$584$	−14.4641	−0.598529
$585$	0	0
$586$	−26.6603	−1.10132
$587$	− 18.2679i	− 0.753999i	−0.926213	−	0.376999i	$-0.876956\pi$
$587$	− 18.2679i	− 0.753999i	0.926213	−	0.376999i	$-0.123044\pi$
$588$	0.464102i	0.0191392i
$589$	5.92820	0.244267
$590$	0	0
$591$	−0.339746	−0.0139753
$592$	− 8.92820i	− 0.366947i
$593$	30.2487i	1.24217i	0.783745	+	0.621083i	$0.213307\pi$
$593$	30.2487i	1.24217i	−0.783745	+	0.621083i	$0.786693\pi$
$594$	5.19615	0.213201
$595$	0	0
$596$	−21.8564	−0.895273
$597$	− 3.07180i	− 0.125720i
$598$	− 40.0526i	− 1.63787i
$599$	16.9808	0.693815	0.346908	−	0.937899i	$-0.387232\pi$
$599$	16.9808	0.693815	0.346908	+	0.937899i	$0.387232\pi$
$600$	0	0
$601$	25.8038	1.05256	0.526280	−	0.850311i	$-0.323586\pi$
$601$	25.8038	1.05256	0.526280	+	0.850311i	$0.323586\pi$
$602$	23.8564i	0.972315i
$603$	5.19615i	0.211604i
$604$	−10.3923	−0.422857
$605$	0	0
$606$	11.6603	0.473665
$607$	− 35.1051i	− 1.42487i	−0.701737	−	0.712436i	$-0.747592\pi$
$607$	− 35.1051i	− 1.42487i	0.701737	−	0.712436i	$-0.252408\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	−25.1244	−1.01809
$610$	0	0
$611$	−16.3923	−0.663162
$612$	− 2.73205i	− 0.110437i
$613$	− 7.07180i	− 0.285627i	−0.989750	−	0.142814i	$-0.954385\pi$
$613$	− 7.07180i	− 0.285627i	0.989750	−	0.142814i	$-0.0456149\pi$
$614$	−4.66025	−0.188073
$615$	0	0
$616$	14.1962	0.571979
$617$	1.32051i	0.0531617i	0.999647	+	0.0265808i	$0.00846194\pi$
$617$	1.32051i	0.0531617i	−0.999647	+	0.0265808i	$0.991538\pi$
$618$	− 14.8564i	− 0.597612i
$619$	−26.9808	−1.08445	−0.542224	−	0.840234i	$-0.682418\pi$
$619$	−26.9808	−1.08445	−0.542224	+	0.840234i	$0.682418\pi$
$620$	0	0
$621$	−8.46410	−0.339653
$622$	− 15.4641i	− 0.620054i
$623$	− 10.7321i	− 0.429971i
$624$	4.73205	0.189434
$625$	0	0
$626$	−4.85641	−0.194101
$627$	− 5.19615i	− 0.207514i
$628$	− 21.8564i	− 0.872166i
$629$	−24.3923	−0.972585
$630$	0	0
$631$	−2.92820	−0.116570	−0.0582850	−	0.998300i	$-0.518563\pi$
$631$	−2.92820	−0.116570	−0.0582850	+	0.998300i	$0.518563\pi$
$632$	5.92820i	0.235811i
$633$	7.19615i	0.286021i
$634$	−23.0526	−0.915534
$635$	0	0
$636$	4.66025	0.184791
$637$	2.19615i	0.0870147i
$638$	47.7846i	1.89181i
$639$	10.1962	0.403354
$640$	0	0
$641$	−12.3923	−0.489467	−0.244733	−	0.969590i	$-0.578700\pi$
$641$	−12.3923	−0.489467	−0.244733	+	0.969590i	$0.578700\pi$
$642$	− 4.19615i	− 0.165609i
$643$	− 32.9282i	− 1.29856i	−0.760549	−	0.649281i	$-0.775070\pi$
$643$	− 32.9282i	− 1.29856i	0.760549	−	0.649281i	$-0.224930\pi$
$644$	−23.1244	−0.911227
$645$	0	0
$646$	−2.73205	−0.107491
$647$	− 13.9282i	− 0.547574i	−0.961790	−	0.273787i	$-0.911724\pi$
$647$	− 13.9282i	− 0.547574i	0.961790	−	0.273787i	$-0.0882764\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−11.4115	−0.447942
$650$	0	0
$651$	16.1962	0.634777
$652$	− 10.0000i	− 0.391630i
$653$	− 25.1769i	− 0.985249i	−0.870242	−	0.492624i	$-0.836038\pi$
$653$	− 25.1769i	− 0.985249i	0.870242	−	0.492624i	$-0.163962\pi$
$654$	−10.3923	−0.406371
$655$	0	0
$656$	0	0
$657$	14.4641i	0.564298i
$658$	9.46410i	0.368949i
$659$	7.26795	0.283119	0.141560	−	0.989930i	$-0.454788\pi$
$659$	7.26795	0.283119	0.141560	+	0.989930i	$0.454788\pi$
$660$	0	0
$661$	26.9282	1.04739	0.523693	−	0.851907i	$-0.324554\pi$
$661$	26.9282	1.04739	0.523693	+	0.851907i	$0.324554\pi$
$662$	19.1962i	0.746080i
$663$	− 12.9282i	− 0.502090i
$664$	−16.6603	−0.646543
$665$	0	0
$666$	−8.92820	−0.345961
$667$	− 77.8372i	− 3.01387i
$668$	− 2.00000i	− 0.0773823i
$669$	17.9282	0.693144
$670$	0	0
$671$	42.9615	1.65851
$672$	− 2.73205i	− 0.105391i
$673$	29.2679i	1.12820i	0.825708	+	0.564098i	$0.190776\pi$
$673$	29.2679i	1.12820i	−0.825708	+	0.564098i	$0.809224\pi$
$674$	−15.3205	−0.590124
$675$	0	0
$676$	9.39230	0.361242
$677$	− 23.8756i	− 0.917616i	−0.888536	−	0.458808i	$-0.848277\pi$
$677$	− 23.8756i	− 0.917616i	0.888536	−	0.458808i	$-0.151723\pi$
$678$	− 14.3205i	− 0.549976i
$679$	−46.7846	−1.79543
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	− 30.8038i	− 1.17954i
$683$	20.8756i	0.798784i	0.916780	+	0.399392i	$0.130779\pi$
$683$	20.8756i	0.798784i	−0.916780	+	0.399392i	$0.869221\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−17.8564	−0.681761
$687$	− 5.19615i	− 0.198246i
$688$	− 8.73205i	− 0.332906i
$689$	22.0526	0.840136
$690$	0	0
$691$	19.1244	0.727525	0.363762	−	0.931492i	$-0.381492\pi$
$691$	19.1244	0.727525	0.363762	+	0.931492i	$0.381492\pi$
$692$	22.6603i	0.861414i
$693$	− 14.1962i	− 0.539267i
$694$	−1.07180	−0.0406848
$695$	0	0
$696$	9.19615	0.348579
$697$	0	0
$698$	13.5885i	0.514331i
$699$	−17.6603	−0.667972
$700$	0	0
$701$	43.3205	1.63619	0.818097	−	0.575081i	$-0.195029\pi$
$701$	43.3205	1.63619	0.818097	+	0.575081i	$0.195029\pi$
$702$	− 4.73205i	− 0.178600i
$703$	8.92820i	0.336734i
$704$	−5.19615	−0.195837
$705$	0	0
$706$	−3.41154	−0.128395
$707$	− 31.8564i	− 1.19808i
$708$	2.19615i	0.0825365i
$709$	29.4449	1.10583	0.552913	−	0.833239i	$-0.313516\pi$
$709$	29.4449	1.10583	0.552913	+	0.833239i	$0.313516\pi$
$710$	0	0
$711$	5.92820	0.222325
$712$	3.92820i	0.147216i
$713$	50.1769i	1.87914i
$714$	−7.46410	−0.279337
$715$	0	0
$716$	−9.85641	−0.368351
$717$	7.07180i	0.264101i
$718$	− 19.8564i	− 0.741035i
$719$	40.5692	1.51298	0.756488	−	0.654007i	$-0.226913\pi$
$719$	40.5692	1.51298	0.756488	+	0.654007i	$0.226913\pi$
$720$	0	0
$721$	−40.5885	−1.51159
$722$	1.00000i	0.0372161i
$723$	7.80385i	0.290228i
$724$	−19.2679	−0.716088
$725$	0	0
$726$	−16.0000	−0.593816
$727$	− 3.46410i	− 0.128476i	−0.997935	−	0.0642382i	$-0.979538\pi$
$727$	− 3.46410i	− 0.128476i	0.997935	−	0.0642382i	$-0.0204617\pi$
$728$	− 12.9282i	− 0.479151i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−23.8564	−0.882361
$732$	− 8.26795i	− 0.305592i
$733$	11.7321i	0.433333i	0.976246	+	0.216667i	$0.0695184\pi$
$733$	11.7321i	0.433333i	−0.976246	+	0.216667i	$0.930482\pi$
$734$	−29.5167	−1.08948
$735$	0	0
$736$	8.46410	0.311991
$737$	− 27.0000i	− 0.994558i
$738$	0	0
$739$	22.2487	0.818432	0.409216	−	0.912438i	$-0.365802\pi$
$739$	22.2487	0.818432	0.409216	+	0.912438i	$0.365802\pi$
$740$	0	0
$741$	−4.73205	−0.173836
$742$	− 12.7321i	− 0.467408i
$743$	16.7846i	0.615768i	0.951424	+	0.307884i	$0.0996208\pi$
$743$	16.7846i	0.615768i	−0.951424	+	0.307884i	$0.900379\pi$
$744$	−5.92820	−0.217338
$745$	0	0
$746$	27.6603	1.01271
$747$	16.6603i	0.609567i
$748$	14.1962i	0.519063i
$749$	−11.4641	−0.418889
$750$	0	0
$751$	31.4641	1.14814	0.574071	−	0.818806i	$-0.305363\pi$
$751$	31.4641	1.14814	0.574071	+	0.818806i	$0.305363\pi$
$752$	− 3.46410i	− 0.126323i
$753$	3.46410i	0.126239i
$754$	43.5167	1.58478
$755$	0	0
$756$	−2.73205	−0.0993637
$757$	11.9808i	0.435448i	0.976010	+	0.217724i	$0.0698633\pi$
$757$	11.9808i	0.435448i	−0.976010	+	0.217724i	$0.930137\pi$
$758$	29.1769i	1.05975i
$759$	43.9808	1.59640
$760$	0	0
$761$	−21.5167	−0.779978	−0.389989	−	0.920819i	$-0.627521\pi$
$761$	−21.5167	−0.779978	−0.389989	+	0.920819i	$0.627521\pi$
$762$	9.39230i	0.340247i
$763$	28.3923i	1.02787i
$764$	24.4641	0.885080
$765$	0	0
$766$	4.05256	0.146425
$767$	10.3923i	0.375244i
$768$	1.00000i	0.0360844i
$769$	−51.9282	−1.87258	−0.936289	−	0.351229i	$-0.885764\pi$
$769$	−51.9282	−1.87258	−0.936289	+	0.351229i	$0.885764\pi$
$770$	0	0
$771$	15.9282	0.573640
$772$	− 10.1962i	− 0.366968i
$773$	42.3923i	1.52475i	0.647138	+	0.762373i	$0.275966\pi$
$773$	42.3923i	1.52475i	−0.647138	+	0.762373i	$0.724034\pi$
$774$	−8.73205	−0.313867
$775$	0	0
$776$	17.1244	0.614729
$777$	24.3923i	0.875069i
$778$	− 12.9282i	− 0.463499i
$779$	0	0
$780$	0	0
$781$	−52.9808	−1.89580
$782$	− 23.1244i	− 0.826925i
$783$	− 9.19615i	− 0.328644i
$784$	−0.464102	−0.0165751
$785$	0	0
$786$	11.1962	0.399354
$787$	41.9808i	1.49645i	0.663444	+	0.748226i	$0.269094\pi$
$787$	41.9808i	1.49645i	−0.663444	+	0.748226i	$0.730906\pi$
$788$	− 0.339746i	− 0.0121029i
$789$	12.4641	0.443734
$790$	0	0
$791$	−39.1244	−1.39110
$792$	5.19615i	0.184637i
$793$	− 39.1244i	− 1.38935i
$794$	34.1244	1.21103
$795$	0	0
$796$	3.07180	0.108877
$797$	29.5692i	1.04740i	0.851904	+	0.523698i	$0.175448\pi$
$797$	29.5692i	1.04740i	−0.851904	+	0.523698i	$0.824552\pi$
$798$	2.73205i	0.0967136i
$799$	−9.46410	−0.334816
$800$	0	0
$801$	3.92820	0.138796
$802$	18.7128i	0.660772i
$803$	− 75.1577i	− 2.65226i
$804$	−5.19615	−0.183254
$805$	0	0
$806$	−28.0526	−0.988109
$807$	− 15.4641i	− 0.544362i
$808$	11.6603i	0.410206i
$809$	19.1244	0.672377	0.336188	−	0.941795i	$-0.390862\pi$
$809$	19.1244	0.672377	0.336188	+	0.941795i	$0.390862\pi$
$810$	0	0
$811$	−18.5167	−0.650208	−0.325104	−	0.945678i	$-0.605399\pi$
$811$	−18.5167	−0.650208	−0.325104	+	0.945678i	$0.605399\pi$
$812$	− 25.1244i	− 0.881692i
$813$	− 25.1244i	− 0.881150i
$814$	46.3923	1.62605
$815$	0	0
$816$	2.73205	0.0956409
$817$	8.73205i	0.305496i
$818$	− 17.3205i	− 0.605597i
$819$	−12.9282	−0.451748
$820$	0	0
$821$	36.1962	1.26325	0.631627	−	0.775272i	$-0.282387\pi$
$821$	36.1962	1.26325	0.631627	+	0.775272i	$0.282387\pi$
$822$	− 16.3923i	− 0.571747i
$823$	25.7128i	0.896292i	0.893960	+	0.448146i	$0.147916\pi$
$823$	25.7128i	0.896292i	−0.893960	+	0.448146i	$0.852084\pi$
$824$	14.8564	0.517547
$825$	0	0
$826$	6.00000	0.208767
$827$	12.7321i	0.442737i	0.975190	+	0.221368i	$0.0710523\pi$
$827$	12.7321i	0.442737i	−0.975190	+	0.221368i	$0.928948\pi$
$828$	− 8.46410i	− 0.294148i
$829$	6.73205	0.233814	0.116907	−	0.993143i	$-0.462702\pi$
$829$	6.73205	0.233814	0.116907	+	0.993143i	$0.462702\pi$
$830$	0	0
$831$	24.5167	0.850474
$832$	4.73205i	0.164054i
$833$	1.26795i	0.0439318i
$834$	−3.80385	−0.131716
$835$	0	0
$836$	5.19615	0.179713
$837$	5.92820i	0.204909i
$838$	5.85641i	0.202306i
$839$	35.9090	1.23972	0.619858	−	0.784714i	$-0.287190\pi$
$839$	35.9090	1.23972	0.619858	+	0.784714i	$0.287190\pi$
$840$	0	0
$841$	55.5692	1.91618
$842$	− 2.00000i	− 0.0689246i
$843$	− 11.0000i	− 0.378860i
$844$	−7.19615	−0.247702
$845$	0	0
$846$	−3.46410	−0.119098
$847$	43.7128i	1.50199i
$848$	4.66025i	0.160034i
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−75.5692	−2.59048
$852$	10.1962i	0.349314i
$853$	11.1769i	0.382690i	0.981523	+	0.191345i	$0.0612850\pi$
$853$	11.1769i	0.382690i	−0.981523	+	0.191345i	$0.938715\pi$
$854$	−22.5885	−0.772961
$855$	0	0
$856$	4.19615	0.143422
$857$	13.8564i	0.473326i	0.971592	+	0.236663i	$0.0760537\pi$
$857$	13.8564i	0.473326i	−0.971592	+	0.236663i	$0.923946\pi$
$858$	24.5885i	0.839436i
$859$	30.7846	1.05036	0.525179	−	0.850992i	$-0.323998\pi$
$859$	30.7846	1.05036	0.525179	+	0.850992i	$0.323998\pi$
$860$	0	0
$861$	0	0
$862$	− 1.60770i	− 0.0547583i
$863$	23.6077i	0.803615i	0.915724	+	0.401808i	$0.131618\pi$
$863$	23.6077i	0.803615i	−0.915724	+	0.401808i	$0.868382\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	5.80385	0.197223
$867$	9.53590i	0.323856i
$868$	16.1962i	0.549733i
$869$	−30.8038	−1.04495
$870$	0	0
$871$	−24.5885	−0.833148
$872$	− 10.3923i	− 0.351928i
$873$	− 17.1244i	− 0.579572i
$874$	−8.46410	−0.286303
$875$	0	0
$876$	−14.4641	−0.488697
$877$	51.4641i	1.73782i	0.494971	+	0.868910i	$0.335179\pi$
$877$	51.4641i	1.73782i	−0.494971	+	0.868910i	$0.664821\pi$
$878$	− 3.39230i	− 0.114485i
$879$	−26.6603	−0.899228
$880$	0	0
$881$	19.6077	0.660600	0.330300	−	0.943876i	$-0.392850\pi$
$881$	19.6077	0.660600	0.330300	+	0.943876i	$0.392850\pi$
$882$	0.464102i	0.0156271i
$883$	− 43.9090i	− 1.47765i	−0.673895	−	0.738827i	$-0.735380\pi$
$883$	− 43.9090i	− 1.47765i	0.673895	−	0.738827i	$-0.264620\pi$
$884$	12.9282	0.434823
$885$	0	0
$886$	−20.5167	−0.689271
$887$	− 1.46410i	− 0.0491597i	−0.999698	−	0.0245799i	$-0.992175\pi$
$887$	− 1.46410i	− 0.0491597i	0.999698	−	0.0245799i	$-0.00782480\pi$
$888$	− 8.92820i	− 0.299611i
$889$	25.6603	0.860617
$890$	0	0
$891$	5.19615	0.174078
$892$	17.9282i	0.600281i
$893$	3.46410i	0.115922i
$894$	−21.8564	−0.730988
$895$	0	0
$896$	2.73205	0.0912714
$897$	− 40.0526i	− 1.33732i
$898$	− 0.464102i	− 0.0154873i
$899$	−54.5167	−1.81823
$900$	0	0
$901$	12.7321	0.424166
$902$	0	0
$903$	23.8564i	0.793891i
$904$	14.3205	0.476293
$905$	0	0
$906$	−10.3923	−0.345261
$907$	6.67949i	0.221789i	0.993832	+	0.110894i	$0.0353715\pi$
$907$	6.67949i	0.221789i	−0.993832	+	0.110894i	$0.964628\pi$
$908$	− 2.00000i	− 0.0663723i
$909$	11.6603	0.386746
$910$	0	0
$911$	−17.6077	−0.583369	−0.291684	−	0.956515i	$-0.594216\pi$
$911$	−17.6077	−0.583369	−0.291684	+	0.956515i	$0.594216\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	− 86.5692i	− 2.86502i
$914$	32.0000	1.05847
$915$	0	0
$916$	5.19615	0.171686
$917$	− 30.5885i	− 1.01012i
$918$	− 2.73205i	− 0.0901711i
$919$	−36.8372	−1.21515	−0.607573	−	0.794264i	$-0.707857\pi$
$919$	−36.8372	−1.21515	−0.607573	+	0.794264i	$0.707857\pi$
$920$	0	0
$921$	−4.66025	−0.153561
$922$	− 1.85641i	− 0.0611375i
$923$	48.2487i	1.58813i
$924$	14.1962	0.467019
$925$	0	0
$926$	11.6077	0.381453
$927$	− 14.8564i	− 0.487948i
$928$	9.19615i	0.301878i
$929$	−34.9282	−1.14596	−0.572979	−	0.819570i	$-0.694212\pi$
$929$	−34.9282	−1.14596	−0.572979	+	0.819570i	$0.694212\pi$
$930$	0	0
$931$	0.464102	0.0152103
$932$	− 17.6603i	− 0.578481i
$933$	− 15.4641i	− 0.506272i
$934$	−16.6603	−0.545140
$935$	0	0
$936$	4.73205	0.154672
$937$	5.21539i	0.170379i	0.996365	+	0.0851897i	$0.0271496\pi$
$937$	5.21539i	0.170379i	−0.996365	+	0.0851897i	$0.972850\pi$
$938$	14.1962i	0.463521i
$939$	−4.85641	−0.158483
$940$	0	0
$941$	−7.73205	−0.252058	−0.126029	−	0.992027i	$-0.540223\pi$
$941$	−7.73205	−0.252058	−0.126029	+	0.992027i	$0.540223\pi$
$942$	− 21.8564i	− 0.712120i
$943$	0	0
$944$	−2.19615	−0.0714787
$945$	0	0
$946$	45.3731	1.47521
$947$	− 29.8564i	− 0.970203i	−0.874458	−	0.485101i	$-0.838783\pi$
$947$	− 29.8564i	− 0.970203i	0.874458	−	0.485101i	$-0.161217\pi$
$948$	5.92820i	0.192539i
$949$	−68.4449	−2.22181
$950$	0	0
$951$	−23.0526	−0.747530
$952$	− 7.46410i	− 0.241913i
$953$	7.39230i	0.239460i	0.992806	+	0.119730i	$0.0382029\pi$
$953$	7.39230i	0.239460i	−0.992806	+	0.119730i	$0.961797\pi$
$954$	4.66025	0.150881
$955$	0	0
$956$	−7.07180	−0.228718
$957$	47.7846i	1.54466i
$958$	16.6077i	0.536570i
$959$	−44.7846	−1.44617
$960$	0	0
$961$	4.14359	0.133664
$962$	− 42.2487i	− 1.36215i
$963$	− 4.19615i	− 0.135219i
$964$	−7.80385	−0.251345
$965$	0	0
$966$	−23.1244	−0.744014
$967$	− 49.0333i	− 1.57681i	−0.615160	−	0.788403i	$-0.710908\pi$
$967$	− 49.0333i	− 1.57681i	0.615160	−	0.788403i	$-0.289092\pi$
$968$	− 16.0000i	− 0.514259i
$969$	−2.73205	−0.0877661
$970$	0	0
$971$	−33.8038	−1.08482	−0.542409	−	0.840115i	$-0.682488\pi$
$971$	−33.8038	−1.08482	−0.542409	+	0.840115i	$0.682488\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	10.3923i	0.333162i
$974$	13.0718	0.418847
$975$	0	0
$976$	8.26795	0.264651
$977$	− 2.53590i	− 0.0811306i	−0.999177	−	0.0405653i	$-0.987084\pi$
$977$	− 2.53590i	− 0.0811306i	0.999177	−	0.0405653i	$-0.0129159\pi$
$978$	− 10.0000i	− 0.319765i
$979$	−20.4115	−0.652356
$980$	0	0
$981$	−10.3923	−0.331801
$982$	− 34.9282i	− 1.11460i
$983$	11.6603i	0.371904i	0.982559	+	0.185952i	$0.0595370\pi$
$983$	11.6603i	0.371904i	−0.982559	+	0.185952i	$0.940463\pi$
$984$	0	0
$985$	0	0
$986$	25.1244	0.800122
$987$	9.46410i	0.301246i
$988$	− 4.73205i	− 0.150547i
$989$	−73.9090	−2.35017
$990$	0	0
$991$	13.7846	0.437883	0.218941	−	0.975738i	$-0.429740\pi$
$991$	13.7846	0.437883	0.218941	+	0.975738i	$0.429740\pi$
$992$	− 5.92820i	− 0.188221i
$993$	19.1962i	0.609171i
$994$	27.8564	0.883552
$995$	0	0
$996$	−16.6603	−0.527900
$997$	23.3397i	0.739177i	0.929196	+	0.369589i	$0.120501\pi$
$997$	23.3397i	0.739177i	−0.929196	+	0.369589i	$0.879499\pi$
$998$	− 13.1244i	− 0.415444i
$999$	−8.92820	−0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.u.799.4		4
5.2	odd	4		2850.2.a.be.1.1	✓	2
5.3	odd	4		2850.2.a.bh.1.2	yes	2
5.4	even	2	inner	2850.2.d.u.799.1		4
15.2	even	4		8550.2.a.bz.1.1		2
15.8	even	4		8550.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.be.1.1	✓	2	5.2	odd	4
2850.2.a.bh.1.2	yes	2	5.3	odd	4
2850.2.d.u.799.1		4	5.4	even	2	inner
2850.2.d.u.799.4		4	1.1	even	1	trivial
8550.2.a.bt.1.2		2	15.8	even	4
8550.2.a.bz.1.1		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.73205i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.73205i q^{7} -1.00000i q^{8} -1.00000 q^{9} +5.19615 q^{11} -1.00000i q^{12} -4.73205i q^{13} -2.73205 q^{14} +1.00000 q^{16} -2.73205i q^{17} -1.00000i q^{18} -1.00000 q^{19} -2.73205 q^{21} +5.19615i q^{22} -8.46410i q^{23} +1.00000 q^{24} +4.73205 q^{26} -1.00000i q^{27} -2.73205i q^{28} +9.19615 q^{29} -5.92820 q^{31} +1.00000i q^{32} +5.19615i q^{33} +2.73205 q^{34} +1.00000 q^{36} -8.92820i q^{37} -1.00000i q^{38} +4.73205 q^{39} -2.73205i q^{42} -8.73205i q^{43} -5.19615 q^{44} +8.46410 q^{46} -3.46410i q^{47} +1.00000i q^{48} -0.464102 q^{49} +2.73205 q^{51} +4.73205i q^{52} +4.66025i q^{53} +1.00000 q^{54} +2.73205 q^{56} -1.00000i q^{57} +9.19615i q^{58} -2.19615 q^{59} +8.26795 q^{61} -5.92820i q^{62} -2.73205i q^{63} -1.00000 q^{64} -5.19615 q^{66} -5.19615i q^{67} +2.73205i q^{68} +8.46410 q^{69} -10.1962 q^{71} +1.00000i q^{72} -14.4641i q^{73} +8.92820 q^{74} +1.00000 q^{76} +14.1962i q^{77} +4.73205i q^{78} -5.92820 q^{79} +1.00000 q^{81} -16.6603i q^{83} +2.73205 q^{84} +8.73205 q^{86} +9.19615i q^{87} -5.19615i q^{88} -3.92820 q^{89} +12.9282 q^{91} +8.46410i q^{92} -5.92820i q^{93} +3.46410 q^{94} -1.00000 q^{96} +17.1244i q^{97} -0.464102i q^{98} -5.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 4 q^{14} + 4 q^{16} - 4 q^{19} - 4 q^{21} + 4 q^{24} + 12 q^{26} + 16 q^{29} + 4 q^{31} + 4 q^{34} + 4 q^{36} + 12 q^{39} + 20 q^{46} + 12 q^{49} + 4 q^{51} + 4 q^{54} + 4 q^{56} + 12 q^{59} + 40 q^{61} - 4 q^{64} + 20 q^{69} - 20 q^{71} + 8 q^{74} + 4 q^{76} + 4 q^{79} + 4 q^{81} + 4 q^{84} + 28 q^{86} + 12 q^{89} + 24 q^{91} - 4 q^{96}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 4 * q^14 + 4 * q^16 - 4 * q^19 - 4 * q^21 + 4 * q^24 + 12 * q^26 + 16 * q^29 + 4 * q^31 + 4 * q^34 + 4 * q^36 + 12 * q^39 + 20 * q^46 + 12 * q^49 + 4 * q^51 + 4 * q^54 + 4 * q^56 + 12 * q^59 + 40 * q^61 - 4 * q^64 + 20 * q^69 - 20 * q^71 + 8 * q^74 + 4 * q^76 + 4 * q^79 + 4 * q^81 + 4 * q^84 + 28 * q^86 + 12 * q^89 + 24 * q^91 - 4 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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