LMFDB - Embedded newform 1110.2.d.f.889.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(889,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.d (of order $2$, degree $1$, 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:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			889.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1110.889
Dual form			1110.2.d.f.889.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.23607 q^{10} +0.763932 q^{11} +1.00000i q^{12} +4.47214i q^{13} -2.00000 q^{14} +2.23607 q^{15} +1.00000 q^{16} +4.47214i q^{17} +1.00000i q^{18} +2.76393 q^{19} -2.23607i q^{20} -2.00000 q^{21} -0.763932i q^{22} +4.00000i q^{23} +1.00000 q^{24} -5.00000 q^{25} +4.47214 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -2.23607i q^{30} +6.47214 q^{31} -1.00000i q^{32} -0.763932i q^{33} +4.47214 q^{34} +4.47214 q^{35} +1.00000 q^{36} -1.00000i q^{37} -2.76393i q^{38} +4.47214 q^{39} -2.23607 q^{40} +4.47214 q^{41} +2.00000i q^{42} +1.52786i q^{43} -0.763932 q^{44} -2.23607i q^{45} +4.00000 q^{46} +3.70820i q^{47} -1.00000i q^{48} +3.00000 q^{49} +5.00000i q^{50} +4.47214 q^{51} -4.47214i q^{52} +3.52786i q^{53} +1.00000 q^{54} +1.70820i q^{55} +2.00000 q^{56} -2.76393i q^{57} +4.00000i q^{58} +10.9443 q^{59} -2.23607 q^{60} +0.763932 q^{61} -6.47214i q^{62} +2.00000i q^{63} -1.00000 q^{64} -10.0000 q^{65} -0.763932 q^{66} +1.52786i q^{67} -4.47214i q^{68} +4.00000 q^{69} -4.47214i q^{70} +6.47214 q^{71} -1.00000i q^{72} +1.52786i q^{73} -1.00000 q^{74} +5.00000i q^{75} -2.76393 q^{76} -1.52786i q^{77} -4.47214i q^{78} -6.47214 q^{79} +2.23607i q^{80} +1.00000 q^{81} -4.47214i q^{82} +11.4164i q^{83} +2.00000 q^{84} -10.0000 q^{85} +1.52786 q^{86} +4.00000i q^{87} +0.763932i q^{88} +16.4721 q^{89} -2.23607 q^{90} +8.94427 q^{91} -4.00000i q^{92} -6.47214i q^{93} +3.70820 q^{94} +6.18034i q^{95} -1.00000 q^{96} -14.1803i q^{97} -3.00000i q^{98} -0.763932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{11} - 8 q^{14} + 4 q^{16} + 20 q^{19} - 8 q^{21} + 4 q^{24} - 20 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{36} - 12 q^{44} + 16 q^{46} + 12 q^{49} + 4 q^{54} + 8 q^{56} + 8 q^{59} + 12 q^{61} - 4 q^{64} - 40 q^{65} - 12 q^{66} + 16 q^{69} + 8 q^{71} - 4 q^{74} - 20 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} - 40 q^{85} + 24 q^{86} + 48 q^{89} - 12 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 12 * q^11 - 8 * q^14 + 4 * q^16 + 20 * q^19 - 8 * q^21 + 4 * q^24 - 20 * q^25 - 16 * q^29 + 8 * q^31 + 4 * q^36 - 12 * q^44 + 16 * q^46 + 12 * q^49 + 4 * q^54 + 8 * q^56 + 8 * q^59 + 12 * q^61 - 4 * q^64 - 40 * q^65 - 12 * q^66 + 16 * q^69 + 8 * q^71 - 4 * q^74 - 20 * q^76 - 8 * q^79 + 4 * q^81 + 8 * q^84 - 40 * q^85 + 24 * q^86 + 48 * q^89 - 12 * q^94 - 4 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	−1.00000	−0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	2.23607	0.707107
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	1.00000i	0.288675i
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	−2.00000	−0.534522
$15$	2.23607	0.577350
$16$	1.00000	0.250000
$17$	4.47214i	1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$	4.47214i	1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$	1.00000i	0.235702i
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	− 2.23607i	− 0.500000i
$21$	−2.00000	−0.436436
$22$	− 0.763932i	− 0.162871i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	4.47214	0.877058
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	− 2.23607i	− 0.408248i
$31$	6.47214	1.16243	0.581215	−	0.813750i	$-0.302578\pi$
$31$	6.47214	1.16243	0.581215	+	0.813750i	$0.302578\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 0.763932i	− 0.132983i
$34$	4.47214	0.766965
$35$	4.47214	0.755929
$36$	1.00000	0.166667
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	− 2.76393i	− 0.448369i
$39$	4.47214	0.716115
$40$	−2.23607	−0.353553
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	2.00000i	0.308607i
$43$	1.52786i	0.232997i	0.993191	+	0.116499i	$0.0371670\pi$
$43$	1.52786i	0.232997i	−0.993191	+	0.116499i	$0.962833\pi$
$44$	−0.763932	−0.115167
$45$	− 2.23607i	− 0.333333i
$46$	4.00000	0.589768
$47$	3.70820i	0.540897i	0.962734	+	0.270449i	$0.0871720\pi$
$47$	3.70820i	0.540897i	−0.962734	+	0.270449i	$0.912828\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	5.00000i	0.707107i
$51$	4.47214	0.626224
$52$	− 4.47214i	− 0.620174i
$53$	3.52786i	0.484589i	0.970203	+	0.242295i	$0.0779001\pi$
$53$	3.52786i	0.484589i	−0.970203	+	0.242295i	$0.922100\pi$
$54$	1.00000	0.136083
$55$	1.70820i	0.230334i
$56$	2.00000	0.267261
$57$	− 2.76393i	− 0.366092i
$58$	4.00000i	0.525226i
$59$	10.9443	1.42482	0.712411	−	0.701762i	$-0.247603\pi$
$59$	10.9443	1.42482	0.712411	+	0.701762i	$0.247603\pi$
$60$	−2.23607	−0.288675
$61$	0.763932	0.0978115	0.0489057	−	0.998803i	$-0.484427\pi$
$61$	0.763932	0.0978115	0.0489057	+	0.998803i	$0.484427\pi$
$62$	− 6.47214i	− 0.821962i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	−10.0000	−1.24035
$66$	−0.763932	−0.0940335
$67$	1.52786i	0.186658i	0.995635	+	0.0933292i	$0.0297509\pi$
$67$	1.52786i	0.186658i	−0.995635	+	0.0933292i	$0.970249\pi$
$68$	− 4.47214i	− 0.542326i
$69$	4.00000	0.481543
$70$	− 4.47214i	− 0.534522i
$71$	6.47214	0.768101	0.384051	−	0.923312i	$-0.374529\pi$
$71$	6.47214	0.768101	0.384051	+	0.923312i	$0.374529\pi$
$72$	− 1.00000i	− 0.117851i
$73$	1.52786i	0.178823i	0.995995	+	0.0894115i	$0.0284986\pi$
$73$	1.52786i	0.178823i	−0.995995	+	0.0894115i	$0.971501\pi$
$74$	−1.00000	−0.116248
$75$	5.00000i	0.577350i
$76$	−2.76393	−0.317045
$77$	− 1.52786i	− 0.174116i
$78$	− 4.47214i	− 0.506370i
$79$	−6.47214	−0.728172	−0.364086	−	0.931365i	$-0.618619\pi$
$79$	−6.47214	−0.728172	−0.364086	+	0.931365i	$0.618619\pi$
$80$	2.23607i	0.250000i
$81$	1.00000	0.111111
$82$	− 4.47214i	− 0.493865i
$83$	11.4164i	1.25311i	0.779376	+	0.626557i	$0.215536\pi$
$83$	11.4164i	1.25311i	−0.779376	+	0.626557i	$0.784464\pi$
$84$	2.00000	0.218218
$85$	−10.0000	−1.08465
$86$	1.52786	0.164754
$87$	4.00000i	0.428845i
$88$	0.763932i	0.0814354i
$89$	16.4721	1.74604	0.873021	−	0.487682i	$-0.162157\pi$
$89$	16.4721	1.74604	0.873021	+	0.487682i	$0.162157\pi$
$90$	−2.23607	−0.235702
$91$	8.94427	0.937614
$92$	− 4.00000i	− 0.417029i
$93$	− 6.47214i	− 0.671129i
$94$	3.70820	0.382472
$95$	6.18034i	0.634089i
$96$	−1.00000	−0.102062
$97$	− 14.1803i	− 1.43980i	−0.694080	−	0.719898i	$-0.744189\pi$
$97$	− 14.1803i	− 1.43980i	0.694080	−	0.719898i	$-0.255811\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−0.763932	−0.0767781
$100$	5.00000	0.500000
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	− 4.47214i	− 0.442807i
$103$	− 6.29180i	− 0.619949i	−0.950745	−	0.309975i	$-0.899679\pi$
$103$	− 6.29180i	− 0.619949i	0.950745	−	0.309975i	$-0.100321\pi$
$104$	−4.47214	−0.438529
$105$	− 4.47214i	− 0.436436i
$106$	3.52786	0.342656
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−8.18034	−0.783534	−0.391767	−	0.920064i	$-0.628136\pi$
$109$	−8.18034	−0.783534	−0.391767	+	0.920064i	$0.628136\pi$
$110$	1.70820	0.162871
$111$	−1.00000	−0.0949158
$112$	− 2.00000i	− 0.188982i
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	−2.76393	−0.258866
$115$	−8.94427	−0.834058
$116$	4.00000	0.371391
$117$	− 4.47214i	− 0.413449i
$118$	− 10.9443i	− 1.00750i
$119$	8.94427	0.819920
$120$	2.23607i	0.204124i
$121$	−10.4164	−0.946946
$122$	− 0.763932i	− 0.0691632i
$123$	− 4.47214i	− 0.403239i
$124$	−6.47214	−0.581215
$125$	− 11.1803i	− 1.00000i
$126$	2.00000	0.178174
$127$	− 14.9443i	− 1.32609i	−0.748580	−	0.663045i	$-0.769264\pi$
$127$	− 14.9443i	− 1.32609i	0.748580	−	0.663045i	$-0.230736\pi$
$128$	1.00000i	0.0883883i
$129$	1.52786	0.134521
$130$	10.0000i	0.877058i
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	0.763932i	0.0664917i
$133$	− 5.52786i	− 0.479327i
$134$	1.52786	0.131987
$135$	−2.23607	−0.192450
$136$	−4.47214	−0.383482
$137$	5.70820i	0.487685i	0.969815	+	0.243842i	$0.0784080\pi$
$137$	5.70820i	0.487685i	−0.969815	+	0.243842i	$0.921592\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−4.47214	−0.377964
$141$	3.70820	0.312287
$142$	− 6.47214i	− 0.543130i
$143$	3.41641i	0.285694i
$144$	−1.00000	−0.0833333
$145$	− 8.94427i	− 0.742781i
$146$	1.52786	0.126447
$147$	− 3.00000i	− 0.247436i
$148$	1.00000i	0.0821995i
$149$	12.6525	1.03653	0.518266	−	0.855220i	$-0.326578\pi$
$149$	12.6525	1.03653	0.518266	+	0.855220i	$0.326578\pi$
$150$	5.00000	0.408248
$151$	−8.94427	−0.727875	−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427	−0.727875	−0.363937	+	0.931423i	$0.618568\pi$
$152$	2.76393i	0.224184i
$153$	− 4.47214i	− 0.361551i
$154$	−1.52786	−0.123119
$155$	14.4721i	1.16243i
$156$	−4.47214	−0.358057
$157$	2.94427i	0.234978i	0.993074	+	0.117489i	$0.0374846\pi$
$157$	2.94427i	0.234978i	−0.993074	+	0.117489i	$0.962515\pi$
$158$	6.47214i	0.514895i
$159$	3.52786	0.279778
$160$	2.23607	0.176777
$161$	8.00000	0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	3.05573i	0.239343i	0.992814	+	0.119672i	$0.0381842\pi$
$163$	3.05573i	0.239343i	−0.992814	+	0.119672i	$0.961816\pi$
$164$	−4.47214	−0.349215
$165$	1.70820	0.132983
$166$	11.4164	0.886085
$167$	6.47214i	0.500829i	0.968139	+	0.250414i	$0.0805669\pi$
$167$	6.47214i	0.500829i	−0.968139	+	0.250414i	$0.919433\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−7.00000	−0.538462
$170$	10.0000i	0.766965i
$171$	−2.76393	−0.211363
$172$	− 1.52786i	− 0.116499i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	4.00000	0.303239
$175$	10.0000i	0.755929i
$176$	0.763932	0.0575835
$177$	− 10.9443i	− 0.822622i
$178$	− 16.4721i	− 1.23464i
$179$	1.05573	0.0789088	0.0394544	−	0.999221i	$-0.487438\pi$
$179$	1.05573	0.0789088	0.0394544	+	0.999221i	$0.487438\pi$
$180$	2.23607i	0.166667i
$181$	−10.9443	−0.813481	−0.406741	−	0.913544i	$-0.633335\pi$
$181$	−10.9443	−0.813481	−0.406741	+	0.913544i	$0.633335\pi$
$182$	− 8.94427i	− 0.662994i
$183$	− 0.763932i	− 0.0564715i
$184$	−4.00000	−0.294884
$185$	2.23607	0.164399
$186$	−6.47214	−0.474560
$187$	3.41641i	0.249832i
$188$	− 3.70820i	− 0.270449i
$189$	2.00000	0.145479
$190$	6.18034	0.448369
$191$	−23.4164	−1.69435	−0.847176	−	0.531313i	$-0.821699\pi$
$191$	−23.4164	−1.69435	−0.847176	+	0.531313i	$0.821699\pi$
$192$	1.00000i	0.0721688i
$193$	18.1803i	1.30865i	0.756214	+	0.654325i	$0.227047\pi$
$193$	18.1803i	1.30865i	−0.756214	+	0.654325i	$0.772953\pi$
$194$	−14.1803	−1.01809
$195$	10.0000i	0.716115i
$196$	−3.00000	−0.214286
$197$	6.94427i	0.494759i	0.968919	+	0.247379i	$0.0795694\pi$
$197$	6.94427i	0.494759i	−0.968919	+	0.247379i	$0.920431\pi$
$198$	0.763932i	0.0542903i
$199$	14.4721	1.02590	0.512951	−	0.858418i	$-0.328552\pi$
$199$	14.4721	1.02590	0.512951	+	0.858418i	$0.328552\pi$
$200$	− 5.00000i	− 0.353553i
$201$	1.52786	0.107767
$202$	9.23607i	0.649847i
$203$	8.00000i	0.561490i
$204$	−4.47214	−0.313112
$205$	10.0000i	0.698430i
$206$	−6.29180	−0.438370
$207$	− 4.00000i	− 0.278019i
$208$	4.47214i	0.310087i
$209$	2.11146	0.146052
$210$	−4.47214	−0.308607
$211$	17.5279	1.20667	0.603334	−	0.797489i	$-0.293839\pi$
$211$	17.5279	1.20667	0.603334	+	0.797489i	$0.293839\pi$
$212$	− 3.52786i	− 0.242295i
$213$	− 6.47214i	− 0.443463i
$214$	−4.00000	−0.273434
$215$	−3.41641	−0.232997
$216$	−1.00000	−0.0680414
$217$	− 12.9443i	− 0.878714i
$218$	8.18034i	0.554043i
$219$	1.52786	0.103243
$220$	− 1.70820i	− 0.115167i
$221$	−20.0000	−1.34535
$222$	1.00000i	0.0671156i
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	−2.00000	−0.133631
$225$	5.00000	0.333333
$226$	10.0000	0.665190
$227$	− 11.0557i	− 0.733794i	−0.930261	−	0.366897i	$-0.880420\pi$
$227$	− 11.0557i	− 0.733794i	0.930261	−	0.366897i	$-0.119580\pi$
$228$	2.76393i	0.183046i
$229$	−13.4164	−0.886581	−0.443291	−	0.896378i	$-0.646189\pi$
$229$	−13.4164	−0.886581	−0.443291	+	0.896378i	$0.646189\pi$
$230$	8.94427i	0.589768i
$231$	−1.52786	−0.100526
$232$	− 4.00000i	− 0.262613i
$233$	13.7082i	0.898054i	0.893518	+	0.449027i	$0.148229\pi$
$233$	13.7082i	0.898054i	−0.893518	+	0.449027i	$0.851771\pi$
$234$	−4.47214	−0.292353
$235$	−8.29180	−0.540897
$236$	−10.9443	−0.712411
$237$	6.47214i	0.420410i
$238$	− 8.94427i	− 0.579771i
$239$	−2.47214	−0.159909	−0.0799546	−	0.996799i	$-0.525478\pi$
$239$	−2.47214	−0.159909	−0.0799546	+	0.996799i	$0.525478\pi$
$240$	2.23607	0.144338
$241$	3.52786	0.227250	0.113625	−	0.993524i	$-0.463754\pi$
$241$	3.52786	0.227250	0.113625	+	0.993524i	$0.463754\pi$
$242$	10.4164i	0.669592i
$243$	− 1.00000i	− 0.0641500i
$244$	−0.763932	−0.0489057
$245$	6.70820i	0.428571i
$246$	−4.47214	−0.285133
$247$	12.3607i	0.786491i
$248$	6.47214i	0.410981i
$249$	11.4164	0.723485
$250$	−11.1803	−0.707107
$251$	−12.4721	−0.787234	−0.393617	−	0.919274i	$-0.628776\pi$
$251$	−12.4721	−0.787234	−0.393617	+	0.919274i	$0.628776\pi$
$252$	− 2.00000i	− 0.125988i
$253$	3.05573i	0.192112i
$254$	−14.9443	−0.937687
$255$	10.0000i	0.626224i
$256$	1.00000	0.0625000
$257$	− 22.9443i	− 1.43122i	−0.698498	−	0.715612i	$-0.746148\pi$
$257$	− 22.9443i	− 1.43122i	0.698498	−	0.715612i	$-0.253852\pi$
$258$	− 1.52786i	− 0.0951207i
$259$	−2.00000	−0.124274
$260$	10.0000	0.620174
$261$	4.00000	0.247594
$262$	2.00000i	0.123560i
$263$	20.6525i	1.27349i	0.771076	+	0.636743i	$0.219719\pi$
$263$	20.6525i	1.27349i	−0.771076	+	0.636743i	$0.780281\pi$
$264$	0.763932	0.0470168
$265$	−7.88854	−0.484589
$266$	−5.52786	−0.338935
$267$	− 16.4721i	− 1.00808i
$268$	− 1.52786i	− 0.0933292i
$269$	−16.6525	−1.01532	−0.507660	−	0.861558i	$-0.669489\pi$
$269$	−16.6525	−1.01532	−0.507660	+	0.861558i	$0.669489\pi$
$270$	2.23607i	0.136083i
$271$	26.8328	1.62998	0.814989	−	0.579477i	$-0.196743\pi$
$271$	26.8328	1.62998	0.814989	+	0.579477i	$0.196743\pi$
$272$	4.47214i	0.271163i
$273$	− 8.94427i	− 0.541332i
$274$	5.70820	0.344845
$275$	−3.81966	−0.230334
$276$	−4.00000	−0.240772
$277$	− 8.47214i	− 0.509041i	−0.967067	−	0.254521i	$-0.918082\pi$
$277$	− 8.47214i	− 0.509041i	0.967067	−	0.254521i	$-0.0819177\pi$
$278$	20.0000i	1.19952i
$279$	−6.47214	−0.387477
$280$	4.47214i	0.267261i
$281$	26.3607	1.57255	0.786273	−	0.617879i	$-0.212008\pi$
$281$	26.3607	1.57255	0.786273	+	0.617879i	$0.212008\pi$
$282$	− 3.70820i	− 0.220820i
$283$	3.05573i	0.181644i	0.995867	+	0.0908221i	$0.0289495\pi$
$283$	3.05573i	0.181644i	−0.995867	+	0.0908221i	$0.971051\pi$
$284$	−6.47214	−0.384051
$285$	6.18034	0.366092
$286$	3.41641	0.202016
$287$	− 8.94427i	− 0.527964i
$288$	1.00000i	0.0589256i
$289$	−3.00000	−0.176471
$290$	−8.94427	−0.525226
$291$	−14.1803	−0.831266
$292$	− 1.52786i	− 0.0894115i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	−3.00000	−0.174964
$295$	24.4721i	1.42482i
$296$	1.00000	0.0581238
$297$	0.763932i	0.0443278i
$298$	− 12.6525i	− 0.732938i
$299$	−17.8885	−1.03452
$300$	− 5.00000i	− 0.288675i
$301$	3.05573	0.176129
$302$	8.94427i	0.514685i
$303$	9.23607i	0.530598i
$304$	2.76393	0.158522
$305$	1.70820i	0.0978115i
$306$	−4.47214	−0.255655
$307$	− 0.583592i	− 0.0333074i	−0.999861	−	0.0166537i	$-0.994699\pi$
$307$	− 0.583592i	− 0.0333074i	0.999861	−	0.0166537i	$-0.00530128\pi$
$308$	1.52786i	0.0870581i
$309$	−6.29180	−0.357928
$310$	14.4721	0.821962
$311$	−8.94427	−0.507183	−0.253592	−	0.967311i	$-0.581612\pi$
$311$	−8.94427	−0.507183	−0.253592	+	0.967311i	$0.581612\pi$
$312$	4.47214i	0.253185i
$313$	− 15.7082i	− 0.887880i	−0.896056	−	0.443940i	$-0.853580\pi$
$313$	− 15.7082i	− 0.887880i	0.896056	−	0.443940i	$-0.146420\pi$
$314$	2.94427	0.166155
$315$	−4.47214	−0.251976
$316$	6.47214	0.364086
$317$	− 14.9443i	− 0.839354i	−0.907674	−	0.419677i	$-0.862143\pi$
$317$	− 14.9443i	− 0.839354i	0.907674	−	0.419677i	$-0.137857\pi$
$318$	− 3.52786i	− 0.197833i
$319$	−3.05573	−0.171088
$320$	− 2.23607i	− 0.125000i
$321$	−4.00000	−0.223258
$322$	− 8.00000i	− 0.445823i
$323$	12.3607i	0.687767i
$324$	−1.00000	−0.0555556
$325$	− 22.3607i	− 1.24035i
$326$	3.05573	0.169241
$327$	8.18034i	0.452374i
$328$	4.47214i	0.246932i
$329$	7.41641	0.408880
$330$	− 1.70820i	− 0.0940335i
$331$	26.7639	1.47108	0.735539	−	0.677482i	$-0.236929\pi$
$331$	26.7639	1.47108	0.735539	+	0.677482i	$0.236929\pi$
$332$	− 11.4164i	− 0.626557i
$333$	1.00000i	0.0547997i
$334$	6.47214	0.354140
$335$	−3.41641	−0.186658
$336$	−2.00000	−0.109109
$337$	− 11.4164i	− 0.621891i	−0.950428	−	0.310946i	$-0.899354\pi$
$337$	− 11.4164i	− 0.621891i	0.950428	−	0.310946i	$-0.100646\pi$
$338$	7.00000i	0.380750i
$339$	10.0000	0.543125
$340$	10.0000	0.542326
$341$	4.94427	0.267747
$342$	2.76393i	0.149456i
$343$	− 20.0000i	− 1.07990i
$344$	−1.52786	−0.0823769
$345$	8.94427i	0.481543i
$346$	18.0000	0.967686
$347$	− 20.9443i	− 1.12435i	−0.827019	−	0.562174i	$-0.809965\pi$
$347$	− 20.9443i	− 1.12435i	0.827019	−	0.562174i	$-0.190035\pi$
$348$	− 4.00000i	− 0.214423i
$349$	−15.5279	−0.831188	−0.415594	−	0.909550i	$-0.636426\pi$
$349$	−15.5279	−0.831188	−0.415594	+	0.909550i	$0.636426\pi$
$350$	10.0000	0.534522
$351$	−4.47214	−0.238705
$352$	− 0.763932i	− 0.0407177i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−10.9443	−0.581681
$355$	14.4721i	0.768101i
$356$	−16.4721	−0.873021
$357$	− 8.94427i	− 0.473381i
$358$	− 1.05573i	− 0.0557970i
$359$	9.88854	0.521897	0.260949	−	0.965353i	$-0.415965\pi$
$359$	9.88854	0.521897	0.260949	+	0.965353i	$0.415965\pi$
$360$	2.23607	0.117851
$361$	−11.3607	−0.597931
$362$	10.9443i	0.575218i
$363$	10.4164i	0.546720i
$364$	−8.94427	−0.468807
$365$	−3.41641	−0.178823
$366$	−0.763932	−0.0399314
$367$	− 16.8328i	− 0.878666i	−0.898324	−	0.439333i	$-0.855215\pi$
$367$	− 16.8328i	− 0.878666i	0.898324	−	0.439333i	$-0.144785\pi$
$368$	4.00000i	0.208514i
$369$	−4.47214	−0.232810
$370$	− 2.23607i	− 0.116248i
$371$	7.05573	0.366315
$372$	6.47214i	0.335565i
$373$	− 14.9443i	− 0.773785i	−0.922125	−	0.386893i	$-0.873548\pi$
$373$	− 14.9443i	− 0.773785i	0.922125	−	0.386893i	$-0.126452\pi$
$374$	3.41641	0.176658
$375$	−11.1803	−0.577350
$376$	−3.70820	−0.191236
$377$	− 17.8885i	− 0.921307i
$378$	− 2.00000i	− 0.102869i
$379$	−21.8885	−1.12434	−0.562169	−	0.827022i	$-0.690033\pi$
$379$	−21.8885	−1.12434	−0.562169	+	0.827022i	$0.690033\pi$
$380$	− 6.18034i	− 0.317045i
$381$	−14.9443	−0.765618
$382$	23.4164i	1.19809i
$383$	− 33.3050i	− 1.70180i	−0.525325	−	0.850902i	$-0.676056\pi$
$383$	− 33.3050i	− 1.70180i	0.525325	−	0.850902i	$-0.323944\pi$
$384$	1.00000	0.0510310
$385$	3.41641	0.174116
$386$	18.1803	0.925355
$387$	− 1.52786i	− 0.0776657i
$388$	14.1803i	0.719898i
$389$	0.583592	0.0295893	0.0147946	−	0.999891i	$-0.495291\pi$
$389$	0.583592	0.0295893	0.0147946	+	0.999891i	$0.495291\pi$
$390$	10.0000	0.506370
$391$	−17.8885	−0.904663
$392$	3.00000i	0.151523i
$393$	2.00000i	0.100887i
$394$	6.94427	0.349847
$395$	− 14.4721i	− 0.728172i
$396$	0.763932	0.0383890
$397$	31.8885i	1.60044i	0.599706	+	0.800220i	$0.295284\pi$
$397$	31.8885i	1.60044i	−0.599706	+	0.800220i	$0.704716\pi$
$398$	− 14.4721i	− 0.725423i
$399$	−5.52786	−0.276739
$400$	−5.00000	−0.250000
$401$	−36.4721	−1.82133	−0.910666	−	0.413144i	$-0.864431\pi$
$401$	−36.4721	−1.82133	−0.910666	+	0.413144i	$0.864431\pi$
$402$	− 1.52786i	− 0.0762029i
$403$	28.9443i	1.44182i
$404$	9.23607	0.459512
$405$	2.23607i	0.111111i
$406$	8.00000	0.397033
$407$	− 0.763932i	− 0.0378667i
$408$	4.47214i	0.221404i
$409$	35.8885	1.77457	0.887287	−	0.461217i	$-0.152587\pi$
$409$	35.8885	1.77457	0.887287	+	0.461217i	$0.152587\pi$
$410$	10.0000	0.493865
$411$	5.70820	0.281565
$412$	6.29180i	0.309975i
$413$	− 21.8885i	− 1.07706i
$414$	−4.00000	−0.196589
$415$	−25.5279	−1.25311
$416$	4.47214	0.219265
$417$	20.0000i	0.979404i
$418$	− 2.11146i	− 0.103275i
$419$	11.2361	0.548918	0.274459	−	0.961599i	$-0.411501\pi$
$419$	11.2361	0.548918	0.274459	+	0.961599i	$0.411501\pi$
$420$	4.47214i	0.218218i
$421$	1.70820	0.0832528	0.0416264	−	0.999133i	$-0.486746\pi$
$421$	1.70820	0.0832528	0.0416264	+	0.999133i	$0.486746\pi$
$422$	− 17.5279i	− 0.853243i
$423$	− 3.70820i	− 0.180299i
$424$	−3.52786	−0.171328
$425$	− 22.3607i	− 1.08465i
$426$	−6.47214	−0.313576
$427$	− 1.52786i	− 0.0739385i
$428$	4.00000i	0.193347i
$429$	3.41641	0.164946
$430$	3.41641i	0.164754i
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	1.00000i	0.0481125i
$433$	− 30.4721i	− 1.46440i	−0.681091	−	0.732199i	$-0.738494\pi$
$433$	− 30.4721i	− 1.46440i	0.681091	−	0.732199i	$-0.261506\pi$
$434$	−12.9443	−0.621345
$435$	−8.94427	−0.428845
$436$	8.18034	0.391767
$437$	11.0557i	0.528867i
$438$	− 1.52786i	− 0.0730042i
$439$	1.52786	0.0729210	0.0364605	−	0.999335i	$-0.488392\pi$
$439$	1.52786	0.0729210	0.0364605	+	0.999335i	$0.488392\pi$
$440$	−1.70820	−0.0814354
$441$	−3.00000	−0.142857
$442$	20.0000i	0.951303i
$443$	− 26.8328i	− 1.27487i	−0.770506	−	0.637433i	$-0.779996\pi$
$443$	− 26.8328i	− 1.27487i	0.770506	−	0.637433i	$-0.220004\pi$
$444$	1.00000	0.0474579
$445$	36.8328i	1.74604i
$446$	10.0000	0.473514
$447$	− 12.6525i	− 0.598442i
$448$	2.00000i	0.0944911i
$449$	16.4721	0.777368	0.388684	−	0.921371i	$-0.372930\pi$
$449$	16.4721	0.777368	0.388684	+	0.921371i	$0.372930\pi$
$450$	− 5.00000i	− 0.235702i
$451$	3.41641	0.160872
$452$	− 10.0000i	− 0.470360i
$453$	8.94427i	0.420239i
$454$	−11.0557	−0.518871
$455$	20.0000i	0.937614i
$456$	2.76393	0.129433
$457$	− 27.1246i	− 1.26884i	−0.772990	−	0.634418i	$-0.781240\pi$
$457$	− 27.1246i	− 1.26884i	0.772990	−	0.634418i	$-0.218760\pi$
$458$	13.4164i	0.626908i
$459$	−4.47214	−0.208741
$460$	8.94427	0.417029
$461$	−2.47214	−0.115139	−0.0575694	−	0.998342i	$-0.518335\pi$
$461$	−2.47214	−0.115139	−0.0575694	+	0.998342i	$0.518335\pi$
$462$	1.52786i	0.0710827i
$463$	− 23.2361i	− 1.07987i	−0.841706	−	0.539936i	$-0.818448\pi$
$463$	− 23.2361i	− 1.07987i	0.841706	−	0.539936i	$-0.181552\pi$
$464$	−4.00000	−0.185695
$465$	14.4721	0.671129
$466$	13.7082	0.635020
$467$	20.9443i	0.969185i	0.874740	+	0.484593i	$0.161032\pi$
$467$	20.9443i	0.969185i	−0.874740	+	0.484593i	$0.838968\pi$
$468$	4.47214i	0.206725i
$469$	3.05573	0.141100
$470$	8.29180i	0.382472i
$471$	2.94427	0.135665
$472$	10.9443i	0.503751i
$473$	1.16718i	0.0536672i
$474$	6.47214	0.297275
$475$	−13.8197	−0.634089
$476$	−8.94427	−0.409960
$477$	− 3.52786i	− 0.161530i
$478$	2.47214i	0.113073i
$479$	−21.5279	−0.983633	−0.491817	−	0.870699i	$-0.663667\pi$
$479$	−21.5279	−0.983633	−0.491817	+	0.870699i	$0.663667\pi$
$480$	− 2.23607i	− 0.102062i
$481$	4.47214	0.203912
$482$	− 3.52786i	− 0.160690i
$483$	− 8.00000i	− 0.364013i
$484$	10.4164	0.473473
$485$	31.7082	1.43980
$486$	−1.00000	−0.0453609
$487$	9.70820i	0.439921i	0.975509	+	0.219960i	$0.0705928\pi$
$487$	9.70820i	0.439921i	−0.975509	+	0.219960i	$0.929407\pi$
$488$	0.763932i	0.0345816i
$489$	3.05573	0.138185
$490$	6.70820	0.303046
$491$	−16.7639	−0.756546	−0.378273	−	0.925694i	$-0.623482\pi$
$491$	−16.7639	−0.756546	−0.378273	+	0.925694i	$0.623482\pi$
$492$	4.47214i	0.201619i
$493$	− 17.8885i	− 0.805659i
$494$	12.3607	0.556133
$495$	− 1.70820i	− 0.0767781i
$496$	6.47214	0.290607
$497$	− 12.9443i	− 0.580630i
$498$	− 11.4164i	− 0.511581i
$499$	18.1803	0.813864	0.406932	−	0.913459i	$-0.366599\pi$
$499$	18.1803	0.813864	0.406932	+	0.913459i	$0.366599\pi$
$500$	11.1803i	0.500000i
$501$	6.47214	0.289154
$502$	12.4721i	0.556659i
$503$	12.9443i	0.577157i	0.957456	+	0.288578i	$0.0931825\pi$
$503$	12.9443i	0.577157i	−0.957456	+	0.288578i	$0.906817\pi$
$504$	−2.00000	−0.0890871
$505$	− 20.6525i	− 0.919023i
$506$	3.05573	0.135844
$507$	7.00000i	0.310881i
$508$	14.9443i	0.663045i
$509$	27.1246	1.20228	0.601139	−	0.799145i	$-0.294714\pi$
$509$	27.1246	1.20228	0.601139	+	0.799145i	$0.294714\pi$
$510$	10.0000	0.442807
$511$	3.05573	0.135177
$512$	− 1.00000i	− 0.0441942i
$513$	2.76393i	0.122031i
$514$	−22.9443	−1.01203
$515$	14.0689	0.619949
$516$	−1.52786	−0.0672605
$517$	2.83282i	0.124587i
$518$	2.00000i	0.0878750i
$519$	18.0000	0.790112
$520$	− 10.0000i	− 0.438529i
$521$	−15.5279	−0.680288	−0.340144	−	0.940373i	$-0.610476\pi$
$521$	−15.5279	−0.680288	−0.340144	+	0.940373i	$0.610476\pi$
$522$	− 4.00000i	− 0.175075i
$523$	30.8328i	1.34822i	0.738629	+	0.674112i	$0.235474\pi$
$523$	30.8328i	1.34822i	−0.738629	+	0.674112i	$0.764526\pi$
$524$	2.00000	0.0873704
$525$	10.0000	0.436436
$526$	20.6525	0.900491
$527$	28.9443i	1.26083i
$528$	− 0.763932i	− 0.0332459i
$529$	7.00000	0.304348
$530$	7.88854i	0.342656i
$531$	−10.9443	−0.474941
$532$	5.52786i	0.239663i
$533$	20.0000i	0.866296i
$534$	−16.4721	−0.712819
$535$	8.94427	0.386695
$536$	−1.52786	−0.0659937
$537$	− 1.05573i	− 0.0455580i
$538$	16.6525i	0.717939i
$539$	2.29180	0.0987146
$540$	2.23607	0.0962250
$541$	5.70820	0.245415	0.122707	−	0.992443i	$-0.460842\pi$
$541$	5.70820	0.245415	0.122707	+	0.992443i	$0.460842\pi$
$542$	− 26.8328i	− 1.15257i
$543$	10.9443i	0.469664i
$544$	4.47214	0.191741
$545$	− 18.2918i	− 0.783534i
$546$	−8.94427	−0.382780
$547$	− 32.3607i	− 1.38364i	−0.722069	−	0.691821i	$-0.756809\pi$
$547$	− 32.3607i	− 1.38364i	0.722069	−	0.691821i	$-0.243191\pi$
$548$	− 5.70820i	− 0.243842i
$549$	−0.763932	−0.0326038
$550$	3.81966i	0.162871i
$551$	−11.0557	−0.470990
$552$	4.00000i	0.170251i
$553$	12.9443i	0.550446i
$554$	−8.47214	−0.359947
$555$	− 2.23607i	− 0.0949158i
$556$	20.0000	0.848189
$557$	39.8885i	1.69013i	0.534662	+	0.845066i	$0.320439\pi$
$557$	39.8885i	1.69013i	−0.534662	+	0.845066i	$0.679561\pi$
$558$	6.47214i	0.273987i
$559$	−6.83282	−0.288997
$560$	4.47214	0.188982
$561$	3.41641	0.144241
$562$	− 26.3607i	− 1.11196i
$563$	21.8885i	0.922492i	0.887272	+	0.461246i	$0.152597\pi$
$563$	21.8885i	0.922492i	−0.887272	+	0.461246i	$0.847403\pi$
$564$	−3.70820	−0.156144
$565$	−22.3607	−0.940721
$566$	3.05573	0.128442
$567$	− 2.00000i	− 0.0839921i
$568$	6.47214i	0.271565i
$569$	41.4164	1.73627	0.868133	−	0.496332i	$-0.165320\pi$
$569$	41.4164	1.73627	0.868133	+	0.496332i	$0.165320\pi$
$570$	− 6.18034i	− 0.258866i
$571$	2.47214	0.103456	0.0517278	−	0.998661i	$-0.483527\pi$
$571$	2.47214	0.103456	0.0517278	+	0.998661i	$0.483527\pi$
$572$	− 3.41641i	− 0.142847i
$573$	23.4164i	0.978234i
$574$	−8.94427	−0.373327
$575$	− 20.0000i	− 0.834058i
$576$	1.00000	0.0416667
$577$	− 34.5410i	− 1.43796i	−0.695030	−	0.718981i	$-0.744609\pi$
$577$	− 34.5410i	− 1.43796i	0.695030	−	0.718981i	$-0.255391\pi$
$578$	3.00000i	0.124784i
$579$	18.1803	0.755549
$580$	8.94427i	0.371391i
$581$	22.8328	0.947265
$582$	14.1803i	0.587794i
$583$	2.69505i	0.111617i
$584$	−1.52786	−0.0632235
$585$	10.0000	0.413449
$586$	−26.0000	−1.07405
$587$	− 18.8328i	− 0.777313i	−0.921383	−	0.388657i	$-0.872939\pi$
$587$	− 18.8328i	− 0.777313i	0.921383	−	0.388657i	$-0.127061\pi$
$588$	3.00000i	0.123718i
$589$	17.8885	0.737085
$590$	24.4721	1.00750
$591$	6.94427	0.285649
$592$	− 1.00000i	− 0.0410997i
$593$	− 36.5410i	− 1.50056i	−0.661120	−	0.750280i	$-0.729919\pi$
$593$	− 36.5410i	− 1.50056i	0.661120	−	0.750280i	$-0.270081\pi$
$594$	0.763932	0.0313445
$595$	20.0000i	0.819920i
$596$	−12.6525	−0.518266
$597$	− 14.4721i	− 0.592305i
$598$	17.8885i	0.731517i
$599$	−9.88854	−0.404035	−0.202017	−	0.979382i	$-0.564750\pi$
$599$	−9.88854	−0.404035	−0.202017	+	0.979382i	$0.564750\pi$
$600$	−5.00000	−0.204124
$601$	−14.3607	−0.585784	−0.292892	−	0.956145i	$-0.594618\pi$
$601$	−14.3607	−0.585784	−0.292892	+	0.956145i	$0.594618\pi$
$602$	− 3.05573i	− 0.124542i
$603$	− 1.52786i	− 0.0622194i
$604$	8.94427	0.363937
$605$	− 23.2918i	− 0.946946i
$606$	9.23607	0.375190
$607$	12.1803i	0.494385i	0.968966	+	0.247192i	$0.0795080\pi$
$607$	12.1803i	0.494385i	−0.968966	+	0.247192i	$0.920492\pi$
$608$	− 2.76393i	− 0.112092i
$609$	8.00000	0.324176
$610$	1.70820	0.0691632
$611$	−16.5836	−0.670900
$612$	4.47214i	0.180775i
$613$	− 9.05573i	− 0.365757i	−0.983135	−	0.182879i	$-0.941458\pi$
$613$	− 9.05573i	− 0.365757i	0.983135	−	0.182879i	$-0.0585416\pi$
$614$	−0.583592	−0.0235519
$615$	10.0000	0.403239
$616$	1.52786	0.0615594
$617$	− 24.1803i	− 0.973464i	−0.873551	−	0.486732i	$-0.838189\pi$
$617$	− 24.1803i	− 0.973464i	0.873551	−	0.486732i	$-0.161811\pi$
$618$	6.29180i	0.253093i
$619$	32.3607	1.30069	0.650343	−	0.759641i	$-0.274625\pi$
$619$	32.3607	1.30069	0.650343	+	0.759641i	$0.274625\pi$
$620$	− 14.4721i	− 0.581215i
$621$	−4.00000	−0.160514
$622$	8.94427i	0.358633i
$623$	− 32.9443i	− 1.31988i
$624$	4.47214	0.179029
$625$	25.0000	1.00000
$626$	−15.7082	−0.627826
$627$	− 2.11146i	− 0.0843234i
$628$	− 2.94427i	− 0.117489i
$629$	4.47214	0.178316
$630$	4.47214i	0.178174i
$631$	13.8885	0.552894	0.276447	−	0.961029i	$-0.410843\pi$
$631$	13.8885	0.552894	0.276447	+	0.961029i	$0.410843\pi$
$632$	− 6.47214i	− 0.257448i
$633$	− 17.5279i	− 0.696670i
$634$	−14.9443	−0.593513
$635$	33.4164	1.32609
$636$	−3.52786	−0.139889
$637$	13.4164i	0.531577i
$638$	3.05573i	0.120977i
$639$	−6.47214	−0.256034
$640$	−2.23607	−0.0883883
$641$	17.4164	0.687907	0.343953	−	0.938987i	$-0.388234\pi$
$641$	17.4164	0.687907	0.343953	+	0.938987i	$0.388234\pi$
$642$	4.00000i	0.157867i
$643$	14.8328i	0.584949i	0.956273	+	0.292475i	$0.0944787\pi$
$643$	14.8328i	0.584949i	−0.956273	+	0.292475i	$0.905521\pi$
$644$	−8.00000	−0.315244
$645$	3.41641i	0.134521i
$646$	12.3607	0.486324
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	1.00000i	0.0392837i
$649$	8.36068	0.328185
$650$	−22.3607	−0.877058
$651$	−12.9443	−0.507326
$652$	− 3.05573i	− 0.119672i
$653$	− 42.7214i	− 1.67182i	−0.548870	−	0.835908i	$-0.684942\pi$
$653$	− 42.7214i	− 1.67182i	0.548870	−	0.835908i	$-0.315058\pi$
$654$	8.18034	0.319877
$655$	− 4.47214i	− 0.174741i
$656$	4.47214	0.174608
$657$	− 1.52786i	− 0.0596077i
$658$	− 7.41641i	− 0.289122i
$659$	−17.7082	−0.689814	−0.344907	−	0.938637i	$-0.612089\pi$
$659$	−17.7082	−0.689814	−0.344907	+	0.938637i	$0.612089\pi$
$660$	−1.70820	−0.0664917
$661$	−44.7639	−1.74112	−0.870558	−	0.492067i	$-0.836242\pi$
$661$	−44.7639	−1.74112	−0.870558	+	0.492067i	$0.836242\pi$
$662$	− 26.7639i	− 1.04021i
$663$	20.0000i	0.776736i
$664$	−11.4164	−0.443043
$665$	12.3607	0.479327
$666$	1.00000	0.0387492
$667$	− 16.0000i	− 0.619522i
$668$	− 6.47214i	− 0.250414i
$669$	10.0000	0.386622
$670$	3.41641i	0.131987i
$671$	0.583592	0.0225293
$672$	2.00000i	0.0771517i
$673$	− 37.8885i	− 1.46050i	−0.683182	−	0.730248i	$-0.739404\pi$
$673$	− 37.8885i	− 1.46050i	0.683182	−	0.730248i	$-0.260596\pi$
$674$	−11.4164	−0.439744
$675$	− 5.00000i	− 0.192450i
$676$	7.00000	0.269231
$677$	− 1.41641i	− 0.0544370i	−0.999630	−	0.0272185i	$-0.991335\pi$
$677$	− 1.41641i	− 0.0544370i	0.999630	−	0.0272185i	$-0.00866498\pi$
$678$	− 10.0000i	− 0.384048i
$679$	−28.3607	−1.08838
$680$	− 10.0000i	− 0.383482i
$681$	−11.0557	−0.423656
$682$	− 4.94427i	− 0.189326i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	2.76393	0.105682
$685$	−12.7639	−0.487685
$686$	−20.0000	−0.763604
$687$	13.4164i	0.511868i
$688$	1.52786i	0.0582493i
$689$	−15.7771	−0.601059
$690$	8.94427	0.340503
$691$	33.3050	1.26698	0.633490	−	0.773751i	$-0.281622\pi$
$691$	33.3050	1.26698	0.633490	+	0.773751i	$0.281622\pi$
$692$	− 18.0000i	− 0.684257i
$693$	1.52786i	0.0580388i
$694$	−20.9443	−0.795034
$695$	− 44.7214i	− 1.69638i
$696$	−4.00000	−0.151620
$697$	20.0000i	0.757554i
$698$	15.5279i	0.587738i
$699$	13.7082	0.518492
$700$	− 10.0000i	− 0.377964i
$701$	35.4164	1.33766	0.668830	−	0.743416i	$-0.266796\pi$
$701$	35.4164	1.33766	0.668830	+	0.743416i	$0.266796\pi$
$702$	4.47214i	0.168790i
$703$	− 2.76393i	− 0.104244i
$704$	−0.763932	−0.0287918
$705$	8.29180i	0.312287i
$706$	14.0000	0.526897
$707$	18.4721i	0.694716i
$708$	10.9443i	0.411311i
$709$	18.6525	0.700508	0.350254	−	0.936655i	$-0.386095\pi$
$709$	18.6525	0.700508	0.350254	+	0.936655i	$0.386095\pi$
$710$	14.4721	0.543130
$711$	6.47214	0.242724
$712$	16.4721i	0.617319i
$713$	25.8885i	0.969534i
$714$	−8.94427	−0.334731
$715$	−7.63932	−0.285694
$716$	−1.05573	−0.0394544
$717$	2.47214i	0.0923236i
$718$	− 9.88854i	− 0.369037i
$719$	38.8328	1.44822	0.724110	−	0.689685i	$-0.242251\pi$
$719$	38.8328	1.44822	0.724110	+	0.689685i	$0.242251\pi$
$720$	− 2.23607i	− 0.0833333i
$721$	−12.5836	−0.468637
$722$	11.3607i	0.422801i
$723$	− 3.52786i	− 0.131203i
$724$	10.9443	0.406741
$725$	20.0000	0.742781
$726$	10.4164	0.386589
$727$	− 5.70820i	− 0.211706i	−0.994382	−	0.105853i	$-0.966243\pi$
$727$	− 5.70820i	− 0.211706i	0.994382	−	0.105853i	$-0.0337572\pi$
$728$	8.94427i	0.331497i
$729$	−1.00000	−0.0370370
$730$	3.41641i	0.126447i
$731$	−6.83282	−0.252721
$732$	0.763932i	0.0282357i
$733$	− 10.9443i	− 0.404236i	−0.979361	−	0.202118i	$-0.935218\pi$
$733$	− 10.9443i	− 0.404236i	0.979361	−	0.202118i	$-0.0647824\pi$
$734$	−16.8328	−0.621311
$735$	6.70820	0.247436
$736$	4.00000	0.147442
$737$	1.16718i	0.0429938i
$738$	4.47214i	0.164622i
$739$	36.3607	1.33755	0.668775	−	0.743465i	$-0.266819\pi$
$739$	36.3607	1.33755	0.668775	+	0.743465i	$0.266819\pi$
$740$	−2.23607	−0.0821995
$741$	12.3607	0.454081
$742$	− 7.05573i	− 0.259024i
$743$	− 15.3475i	− 0.563046i	−0.959554	−	0.281523i	$-0.909160\pi$
$743$	− 15.3475i	− 0.563046i	0.959554	−	0.281523i	$-0.0908396\pi$
$744$	6.47214	0.237280
$745$	28.2918i	1.03653i
$746$	−14.9443	−0.547149
$747$	− 11.4164i	− 0.417705i
$748$	− 3.41641i	− 0.124916i
$749$	−8.00000	−0.292314
$750$	11.1803i	0.408248i
$751$	10.1115	0.368972	0.184486	−	0.982835i	$-0.440938\pi$
$751$	10.1115	0.368972	0.184486	+	0.982835i	$0.440938\pi$
$752$	3.70820i	0.135224i
$753$	12.4721i	0.454510i
$754$	−17.8885	−0.651462
$755$	− 20.0000i	− 0.727875i
$756$	−2.00000	−0.0727393
$757$	17.4164i	0.633010i	0.948591	+	0.316505i	$0.102509\pi$
$757$	17.4164i	0.633010i	−0.948591	+	0.316505i	$0.897491\pi$
$758$	21.8885i	0.795028i
$759$	3.05573	0.110916
$760$	−6.18034	−0.224184
$761$	−10.5836	−0.383655	−0.191827	−	0.981429i	$-0.561441\pi$
$761$	−10.5836	−0.383655	−0.191827	+	0.981429i	$0.561441\pi$
$762$	14.9443i	0.541374i
$763$	16.3607i	0.592296i
$764$	23.4164	0.847176
$765$	10.0000	0.361551
$766$	−33.3050	−1.20336
$767$	48.9443i	1.76728i
$768$	− 1.00000i	− 0.0360844i
$769$	−16.8328	−0.607007	−0.303503	−	0.952830i	$-0.598156\pi$
$769$	−16.8328	−0.607007	−0.303503	+	0.952830i	$0.598156\pi$
$770$	− 3.41641i	− 0.123119i
$771$	−22.9443	−0.826318
$772$	− 18.1803i	− 0.654325i
$773$	32.8328i	1.18091i	0.807069	+	0.590457i	$0.201053\pi$
$773$	32.8328i	1.18091i	−0.807069	+	0.590457i	$0.798947\pi$
$774$	−1.52786	−0.0549179
$775$	−32.3607	−1.16243
$776$	14.1803	0.509045
$777$	2.00000i	0.0717496i
$778$	− 0.583592i	− 0.0209228i
$779$	12.3607	0.442867
$780$	− 10.0000i	− 0.358057i
$781$	4.94427	0.176920
$782$	17.8885i	0.639693i
$783$	− 4.00000i	− 0.142948i
$784$	3.00000	0.107143
$785$	−6.58359	−0.234978
$786$	2.00000	0.0713376
$787$	29.3050i	1.04461i	0.852759	+	0.522304i	$0.174928\pi$
$787$	29.3050i	1.04461i	−0.852759	+	0.522304i	$0.825072\pi$
$788$	− 6.94427i	− 0.247379i
$789$	20.6525	0.735248
$790$	−14.4721	−0.514895
$791$	20.0000	0.711118
$792$	− 0.763932i	− 0.0271451i
$793$	3.41641i	0.121320i
$794$	31.8885	1.13168
$795$	7.88854i	0.279778i
$796$	−14.4721	−0.512951
$797$	19.8885i	0.704488i	0.935908	+	0.352244i	$0.114581\pi$
$797$	19.8885i	0.704488i	−0.935908	+	0.352244i	$0.885419\pi$
$798$	5.52786i	0.195684i
$799$	−16.5836	−0.586685
$800$	5.00000i	0.176777i
$801$	−16.4721	−0.582014
$802$	36.4721i	1.28788i
$803$	1.16718i	0.0411890i
$804$	−1.52786	−0.0538836
$805$	17.8885i	0.630488i
$806$	28.9443	1.01952
$807$	16.6525i	0.586195i
$808$	− 9.23607i	− 0.324924i
$809$	−18.3607	−0.645527	−0.322764	−	0.946480i	$-0.604612\pi$
$809$	−18.3607	−0.645527	−0.322764	+	0.946480i	$0.604612\pi$
$810$	2.23607	0.0785674
$811$	37.8885	1.33045	0.665223	−	0.746644i	$-0.268336\pi$
$811$	37.8885	1.33045	0.665223	+	0.746644i	$0.268336\pi$
$812$	− 8.00000i	− 0.280745i
$813$	− 26.8328i	− 0.941068i
$814$	−0.763932	−0.0267758
$815$	−6.83282	−0.239343
$816$	4.47214	0.156556
$817$	4.22291i	0.147741i
$818$	− 35.8885i	− 1.25481i
$819$	−8.94427	−0.312538
$820$	− 10.0000i	− 0.349215i
$821$	−49.9574	−1.74353	−0.871763	−	0.489928i	$-0.837023\pi$
$821$	−49.9574	−1.74353	−0.871763	+	0.489928i	$0.837023\pi$
$822$	− 5.70820i	− 0.199096i
$823$	55.3050i	1.92781i	0.266247	+	0.963905i	$0.414216\pi$
$823$	55.3050i	1.92781i	−0.266247	+	0.963905i	$0.585784\pi$
$824$	6.29180	0.219185
$825$	3.81966i	0.132983i
$826$	−21.8885	−0.761600
$827$	41.8885i	1.45661i	0.685254	+	0.728304i	$0.259691\pi$
$827$	41.8885i	1.45661i	−0.685254	+	0.728304i	$0.740309\pi$
$828$	4.00000i	0.139010i
$829$	−26.0689	−0.905410	−0.452705	−	0.891660i	$-0.649541\pi$
$829$	−26.0689	−0.905410	−0.452705	+	0.891660i	$0.649541\pi$
$830$	25.5279i	0.886085i
$831$	−8.47214	−0.293895
$832$	− 4.47214i	− 0.155043i
$833$	13.4164i	0.464851i
$834$	20.0000	0.692543
$835$	−14.4721	−0.500829
$836$	−2.11146	−0.0730262
$837$	6.47214i	0.223710i
$838$	− 11.2361i	− 0.388144i
$839$	41.8885	1.44615	0.723077	−	0.690768i	$-0.242727\pi$
$839$	41.8885	1.44615	0.723077	+	0.690768i	$0.242727\pi$
$840$	4.47214	0.154303
$841$	−13.0000	−0.448276
$842$	− 1.70820i	− 0.0588686i
$843$	− 26.3607i	− 0.907910i
$844$	−17.5279	−0.603334
$845$	− 15.6525i	− 0.538462i
$846$	−3.70820	−0.127491
$847$	20.8328i	0.715824i
$848$	3.52786i	0.121147i
$849$	3.05573	0.104872
$850$	−22.3607	−0.766965
$851$	4.00000	0.137118
$852$	6.47214i	0.221732i
$853$	− 4.47214i	− 0.153123i	−0.997065	−	0.0765615i	$-0.975606\pi$
$853$	− 4.47214i	− 0.153123i	0.997065	−	0.0765615i	$-0.0243942\pi$
$854$	−1.52786	−0.0522824
$855$	− 6.18034i	− 0.211363i
$856$	4.00000	0.136717
$857$	− 34.9443i	− 1.19367i	−0.802363	−	0.596837i	$-0.796424\pi$
$857$	− 34.9443i	− 1.19367i	0.802363	−	0.596837i	$-0.203576\pi$
$858$	− 3.41641i	− 0.116634i
$859$	48.0689	1.64009	0.820045	−	0.572300i	$-0.193949\pi$
$859$	48.0689	1.64009	0.820045	+	0.572300i	$0.193949\pi$
$860$	3.41641	0.116499
$861$	−8.94427	−0.304820
$862$	− 17.8885i	− 0.609286i
$863$	− 18.7639i	− 0.638732i	−0.947632	−	0.319366i	$-0.896530\pi$
$863$	− 18.7639i	− 0.638732i	0.947632	−	0.319366i	$-0.103470\pi$
$864$	1.00000	0.0340207
$865$	−40.2492	−1.36851
$866$	−30.4721	−1.03549
$867$	3.00000i	0.101885i
$868$	12.9443i	0.439357i
$869$	−4.94427	−0.167723
$870$	8.94427i	0.303239i
$871$	−6.83282	−0.231521
$872$	− 8.18034i	− 0.277021i
$873$	14.1803i	0.479932i
$874$	11.0557	0.373966
$875$	−22.3607	−0.755929
$876$	−1.52786	−0.0516217
$877$	− 28.8328i	− 0.973615i	−0.873509	−	0.486808i	$-0.838161\pi$
$877$	− 28.8328i	− 0.973615i	0.873509	−	0.486808i	$-0.161839\pi$
$878$	− 1.52786i	− 0.0515629i
$879$	−26.0000	−0.876958
$880$	1.70820i	0.0575835i
$881$	−46.3607	−1.56193	−0.780965	−	0.624574i	$-0.785273\pi$
$881$	−46.3607	−1.56193	−0.780965	+	0.624574i	$0.785273\pi$
$882$	3.00000i	0.101015i
$883$	− 30.8328i	− 1.03761i	−0.854894	−	0.518803i	$-0.826378\pi$
$883$	− 30.8328i	− 1.03761i	0.854894	−	0.518803i	$-0.173622\pi$
$884$	20.0000	0.672673
$885$	24.4721	0.822622
$886$	−26.8328	−0.901466
$887$	− 47.7082i	− 1.60188i	−0.598741	−	0.800942i	$-0.704332\pi$
$887$	− 47.7082i	− 1.60188i	0.598741	−	0.800942i	$-0.295668\pi$
$888$	− 1.00000i	− 0.0335578i
$889$	−29.8885	−1.00243
$890$	36.8328	1.23464
$891$	0.763932	0.0255927
$892$	− 10.0000i	− 0.334825i
$893$	10.2492i	0.342977i
$894$	−12.6525	−0.423162
$895$	2.36068i	0.0789088i
$896$	2.00000	0.0668153
$897$	17.8885i	0.597281i
$898$	− 16.4721i	− 0.549682i
$899$	−25.8885	−0.863431
$900$	−5.00000	−0.166667
$901$	−15.7771	−0.525611
$902$	− 3.41641i	− 0.113754i
$903$	− 3.05573i	− 0.101688i
$904$	−10.0000	−0.332595
$905$	− 24.4721i	− 0.813481i
$906$	8.94427	0.297154
$907$	− 20.5836i	− 0.683467i	−0.939797	−	0.341733i	$-0.888986\pi$
$907$	− 20.5836i	− 0.683467i	0.939797	−	0.341733i	$-0.111014\pi$
$908$	11.0557i	0.366897i
$909$	9.23607	0.306341
$910$	20.0000	0.662994
$911$	−50.2492	−1.66483	−0.832416	−	0.554152i	$-0.813043\pi$
$911$	−50.2492	−1.66483	−0.832416	+	0.554152i	$0.813043\pi$
$912$	− 2.76393i	− 0.0915229i
$913$	8.72136i	0.288635i
$914$	−27.1246	−0.897202
$915$	1.70820	0.0564715
$916$	13.4164	0.443291
$917$	4.00000i	0.132092i
$918$	4.47214i	0.147602i
$919$	−2.83282	−0.0934460	−0.0467230	−	0.998908i	$-0.514878\pi$
$919$	−2.83282	−0.0934460	−0.0467230	+	0.998908i	$0.514878\pi$
$920$	− 8.94427i	− 0.294884i
$921$	−0.583592	−0.0192300
$922$	2.47214i	0.0814155i
$923$	28.9443i	0.952712i
$924$	1.52786	0.0502630
$925$	5.00000i	0.164399i
$926$	−23.2361	−0.763585
$927$	6.29180i	0.206650i
$928$	4.00000i	0.131306i
$929$	−12.1115	−0.397364	−0.198682	−	0.980064i	$-0.563666\pi$
$929$	−12.1115	−0.397364	−0.198682	+	0.980064i	$0.563666\pi$
$930$	− 14.4721i	− 0.474560i
$931$	8.29180	0.271753
$932$	− 13.7082i	− 0.449027i
$933$	8.94427i	0.292822i
$934$	20.9443	0.685318
$935$	−7.63932	−0.249832
$936$	4.47214	0.146176
$937$	24.5836i	0.803111i	0.915835	+	0.401555i	$0.131530\pi$
$937$	24.5836i	0.803111i	−0.915835	+	0.401555i	$0.868470\pi$
$938$	− 3.05573i	− 0.0997731i
$939$	−15.7082	−0.512618
$940$	8.29180	0.270449
$941$	2.76393	0.0901016	0.0450508	−	0.998985i	$-0.485655\pi$
$941$	2.76393	0.0901016	0.0450508	+	0.998985i	$0.485655\pi$
$942$	− 2.94427i	− 0.0959296i
$943$	17.8885i	0.582531i
$944$	10.9443	0.356206
$945$	4.47214i	0.145479i
$946$	1.16718	0.0379484
$947$	29.8885i	0.971247i	0.874168	+	0.485624i	$0.161407\pi$
$947$	29.8885i	0.971247i	−0.874168	+	0.485624i	$0.838593\pi$
$948$	− 6.47214i	− 0.210205i
$949$	−6.83282	−0.221803
$950$	13.8197i	0.448369i
$951$	−14.9443	−0.484601
$952$	8.94427i	0.289886i
$953$	− 5.70820i	− 0.184907i	−0.995717	−	0.0924534i	$-0.970529\pi$
$953$	− 5.70820i	− 0.184907i	0.995717	−	0.0924534i	$-0.0294709\pi$
$954$	−3.52786	−0.114219
$955$	− 52.3607i	− 1.69435i
$956$	2.47214	0.0799546
$957$	3.05573i	0.0987777i
$958$	21.5279i	0.695534i
$959$	11.4164	0.368655
$960$	−2.23607	−0.0721688
$961$	10.8885	0.351243
$962$	− 4.47214i	− 0.144187i
$963$	4.00000i	0.128898i
$964$	−3.52786	−0.113625
$965$	−40.6525	−1.30865
$966$	−8.00000	−0.257396
$967$	54.4296i	1.75034i	0.483818	+	0.875168i	$0.339249\pi$
$967$	54.4296i	1.75034i	−0.483818	+	0.875168i	$0.660751\pi$
$968$	− 10.4164i	− 0.334796i
$969$	12.3607	0.397082
$970$	− 31.7082i	− 1.01809i
$971$	31.2361	1.00241	0.501207	−	0.865328i	$-0.332890\pi$
$971$	31.2361	1.00241	0.501207	+	0.865328i	$0.332890\pi$
$972$	1.00000i	0.0320750i
$973$	40.0000i	1.28234i
$974$	9.70820	0.311071
$975$	−22.3607	−0.716115
$976$	0.763932	0.0244529
$977$	− 18.3607i	− 0.587410i	−0.955896	−	0.293705i	$-0.905112\pi$
$977$	− 18.3607i	− 0.587410i	0.955896	−	0.293705i	$-0.0948884\pi$
$978$	− 3.05573i	− 0.0977114i
$979$	12.5836	0.402173
$980$	− 6.70820i	− 0.214286i
$981$	8.18034	0.261178
$982$	16.7639i	0.534959i
$983$	− 19.1246i	− 0.609980i	−0.952355	−	0.304990i	$-0.901347\pi$
$983$	− 19.1246i	− 0.609980i	0.952355	−	0.304990i	$-0.0986532\pi$
$984$	4.47214	0.142566
$985$	−15.5279	−0.494759
$986$	−17.8885	−0.569687
$987$	− 7.41641i	− 0.236067i
$988$	− 12.3607i	− 0.393246i
$989$	−6.11146	−0.194333
$990$	−1.70820	−0.0542903
$991$	−23.7771	−0.755304	−0.377652	−	0.925948i	$-0.623268\pi$
$991$	−23.7771	−0.755304	−0.377652	+	0.925948i	$0.623268\pi$
$992$	− 6.47214i	− 0.205491i
$993$	− 26.7639i	− 0.849328i
$994$	−12.9443	−0.410567
$995$	32.3607i	1.02590i
$996$	−11.4164	−0.361743
$997$	47.8885i	1.51665i	0.651879	+	0.758323i	$0.273981\pi$
$997$	47.8885i	1.51665i	−0.651879	+	0.758323i	$0.726019\pi$
$998$	− 18.1803i	− 0.575489i
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.d.f.889.2	✓	4
3.2	odd	2		3330.2.d.j.1999.3		4
5.2	odd	4		5550.2.a.bz.1.1		2
5.3	odd	4		5550.2.a.bu.1.1		2
5.4	even	2	inner	1110.2.d.f.889.3	yes	4
15.14	odd	2		3330.2.d.j.1999.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.d.f.889.2	✓	4	1.1	even	1	trivial
1110.2.d.f.889.3	yes	4	5.4	even	2	inner
3330.2.d.j.1999.2		4	15.14	odd	2
3330.2.d.j.1999.3		4	3.2	odd	2
5550.2.a.bu.1.1		2	5.3	odd	4
5550.2.a.bz.1.1		2	5.2	odd	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +2.23607i q^{5} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.23607 q^{10} +0.763932 q^{11} +1.00000i q^{12} +4.47214i q^{13} -2.00000 q^{14} +2.23607 q^{15} +1.00000 q^{16} +4.47214i q^{17} +1.00000i q^{18} +2.76393 q^{19} -2.23607i q^{20} -2.00000 q^{21} -0.763932i q^{22} +4.00000i q^{23} +1.00000 q^{24} -5.00000 q^{25} +4.47214 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -2.23607i q^{30} +6.47214 q^{31} -1.00000i q^{32} -0.763932i q^{33} +4.47214 q^{34} +4.47214 q^{35} +1.00000 q^{36} -1.00000i q^{37} -2.76393i q^{38} +4.47214 q^{39} -2.23607 q^{40} +4.47214 q^{41} +2.00000i q^{42} +1.52786i q^{43} -0.763932 q^{44} -2.23607i q^{45} +4.00000 q^{46} +3.70820i q^{47} -1.00000i q^{48} +3.00000 q^{49} +5.00000i q^{50} +4.47214 q^{51} -4.47214i q^{52} +3.52786i q^{53} +1.00000 q^{54} +1.70820i q^{55} +2.00000 q^{56} -2.76393i q^{57} +4.00000i q^{58} +10.9443 q^{59} -2.23607 q^{60} +0.763932 q^{61} -6.47214i q^{62} +2.00000i q^{63} -1.00000 q^{64} -10.0000 q^{65} -0.763932 q^{66} +1.52786i q^{67} -4.47214i q^{68} +4.00000 q^{69} -4.47214i q^{70} +6.47214 q^{71} -1.00000i q^{72} +1.52786i q^{73} -1.00000 q^{74} +5.00000i q^{75} -2.76393 q^{76} -1.52786i q^{77} -4.47214i q^{78} -6.47214 q^{79} +2.23607i q^{80} +1.00000 q^{81} -4.47214i q^{82} +11.4164i q^{83} +2.00000 q^{84} -10.0000 q^{85} +1.52786 q^{86} +4.00000i q^{87} +0.763932i q^{88} +16.4721 q^{89} -2.23607 q^{90} +8.94427 q^{91} -4.00000i q^{92} -6.47214i q^{93} +3.70820 q^{94} +6.18034i q^{95} -1.00000 q^{96} -14.1803i q^{97} -3.00000i q^{98} -0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 12 q^{11} - 8 q^{14} + 4 q^{16} + 20 q^{19} - 8 q^{21} + 4 q^{24} - 20 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{36} - 12 q^{44} + 16 q^{46} + 12 q^{49} + 4 q^{54} + 8 q^{56} + 8 q^{59} + 12 q^{61} - 4 q^{64} - 40 q^{65} - 12 q^{66} + 16 q^{69} + 8 q^{71} - 4 q^{74} - 20 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} - 40 q^{85} + 24 q^{86} + 48 q^{89} - 12 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 12 * q^11 - 8 * q^14 + 4 * q^16 + 20 * q^19 - 8 * q^21 + 4 * q^24 - 20 * q^25 - 16 * q^29 + 8 * q^31 + 4 * q^36 - 12 * q^44 + 16 * q^46 + 12 * q^49 + 4 * q^54 + 8 * q^56 + 8 * q^59 + 12 * q^61 - 4 * q^64 - 40 * q^65 - 12 * q^66 + 16 * q^69 + 8 * q^71 - 4 * q^74 - 20 * q^76 - 8 * q^79 + 4 * q^81 + 8 * q^84 - 40 * q^85 + 24 * q^86 + 48 * q^89 - 12 * q^94 - 4 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more