LMFDB - Embedded newform 4008.2.a.m.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.a (trivial)

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:	yes
Analytic conductor:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.68837$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.68837 q^{5} +0.722089 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.68837 q^{5} +0.722089 q^{7} +1.00000 q^{9} +4.92380 q^{11} -5.15606 q^{13} +1.68837 q^{15} +4.50170 q^{17} +3.23024 q^{19} +0.722089 q^{21} +9.11854 q^{23} -2.14939 q^{25} +1.00000 q^{27} +1.58248 q^{29} -2.13513 q^{31} +4.92380 q^{33} +1.21916 q^{35} -8.13571 q^{37} -5.15606 q^{39} +5.37711 q^{41} -4.74675 q^{43} +1.68837 q^{45} -3.74876 q^{47} -6.47859 q^{49} +4.50170 q^{51} +1.47444 q^{53} +8.31323 q^{55} +3.23024 q^{57} +3.12417 q^{59} +9.12058 q^{61} +0.722089 q^{63} -8.70535 q^{65} +11.6505 q^{67} +9.11854 q^{69} -13.6974 q^{71} -2.51026 q^{73} -2.14939 q^{75} +3.55542 q^{77} -12.4674 q^{79} +1.00000 q^{81} -2.96218 q^{83} +7.60056 q^{85} +1.58248 q^{87} +7.54127 q^{89} -3.72313 q^{91} -2.13513 q^{93} +5.45386 q^{95} +15.3021 q^{97} +4.92380 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.68837	0.755064	0.377532	−	0.925997i	$-0.376773\pi$
$5$	1.68837	0.755064	0.377532	+	0.925997i	$0.376773\pi$
$6$	0	0
$7$	0.722089	0.272924	0.136462	−	0.990645i	$-0.456427\pi$
$7$	0.722089	0.272924	0.136462	+	0.990645i	$0.456427\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.92380	1.48458	0.742291	−	0.670077i	$-0.233739\pi$
$11$	4.92380	1.48458	0.742291	+	0.670077i	$0.233739\pi$
$12$	0	0
$13$	−5.15606	−1.43003	−0.715017	−	0.699108i	$-0.753581\pi$
$13$	−5.15606	−1.43003	−0.715017	+	0.699108i	$0.753581\pi$
$14$	0	0
$15$	1.68837	0.435936
$16$	0	0
$17$	4.50170	1.09182	0.545911	−	0.837843i	$-0.316184\pi$
$17$	4.50170	1.09182	0.545911	+	0.837843i	$0.316184\pi$
$18$	0	0
$19$	3.23024	0.741069	0.370535	−	0.928819i	$-0.379175\pi$
$19$	3.23024	0.741069	0.370535	+	0.928819i	$0.379175\pi$
$20$	0	0
$21$	0.722089	0.157573
$22$	0	0
$23$	9.11854	1.90135	0.950673	−	0.310194i	$-0.100394\pi$
$23$	9.11854	1.90135	0.950673	+	0.310194i	$0.100394\pi$
$24$	0	0
$25$	−2.14939	−0.429878
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.58248	0.293859	0.146930	−	0.989147i	$-0.453061\pi$
$29$	1.58248	0.293859	0.146930	+	0.989147i	$0.453061\pi$
$30$	0	0
$31$	−2.13513	−0.383480	−0.191740	−	0.981446i	$-0.561413\pi$
$31$	−2.13513	−0.383480	−0.191740	+	0.981446i	$0.561413\pi$
$32$	0	0
$33$	4.92380	0.857124
$34$	0	0
$35$	1.21916	0.206075
$36$	0	0
$37$	−8.13571	−1.33750	−0.668751	−	0.743486i	$-0.733171\pi$
$37$	−8.13571	−1.33750	−0.668751	+	0.743486i	$0.733171\pi$
$38$	0	0
$39$	−5.15606	−0.825630
$40$	0	0
$41$	5.37711	0.839763	0.419882	−	0.907579i	$-0.362072\pi$
$41$	5.37711	0.839763	0.419882	+	0.907579i	$0.362072\pi$
$42$	0	0
$43$	−4.74675	−0.723872	−0.361936	−	0.932203i	$-0.617884\pi$
$43$	−4.74675	−0.723872	−0.361936	+	0.932203i	$0.617884\pi$
$44$	0	0
$45$	1.68837	0.251688
$46$	0	0
$47$	−3.74876	−0.546812	−0.273406	−	0.961899i	$-0.588150\pi$
$47$	−3.74876	−0.546812	−0.273406	+	0.961899i	$0.588150\pi$
$48$	0	0
$49$	−6.47859	−0.925513
$50$	0	0
$51$	4.50170	0.630364
$52$	0	0
$53$	1.47444	0.202530	0.101265	−	0.994859i	$-0.467711\pi$
$53$	1.47444	0.202530	0.101265	+	0.994859i	$0.467711\pi$
$54$	0	0
$55$	8.31323	1.12096
$56$	0	0
$57$	3.23024	0.427856
$58$	0	0
$59$	3.12417	0.406733	0.203366	−	0.979103i	$-0.434812\pi$
$59$	3.12417	0.406733	0.203366	+	0.979103i	$0.434812\pi$
$60$	0	0
$61$	9.12058	1.16777	0.583886	−	0.811836i	$-0.301532\pi$
$61$	9.12058	1.16777	0.583886	+	0.811836i	$0.301532\pi$
$62$	0	0
$63$	0.722089	0.0909747
$64$	0	0
$65$	−8.70535	−1.07977
$66$	0	0
$67$	11.6505	1.42333	0.711667	−	0.702517i	$-0.247941\pi$
$67$	11.6505	1.42333	0.711667	+	0.702517i	$0.247941\pi$
$68$	0	0
$69$	9.11854	1.09774
$70$	0	0
$71$	−13.6974	−1.62559	−0.812793	−	0.582552i	$-0.802054\pi$
$71$	−13.6974	−1.62559	−0.812793	+	0.582552i	$0.802054\pi$
$72$	0	0
$73$	−2.51026	−0.293804	−0.146902	−	0.989151i	$-0.546930\pi$
$73$	−2.51026	−0.293804	−0.146902	+	0.989151i	$0.546930\pi$
$74$	0	0
$75$	−2.14939	−0.248190
$76$	0	0
$77$	3.55542	0.405178
$78$	0	0
$79$	−12.4674	−1.40269	−0.701344	−	0.712823i	$-0.747416\pi$
$79$	−12.4674	−1.40269	−0.701344	+	0.712823i	$0.747416\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.96218	−0.325141	−0.162571	−	0.986697i	$-0.551979\pi$
$83$	−2.96218	−0.325141	−0.162571	+	0.986697i	$0.551979\pi$
$84$	0	0
$85$	7.60056	0.824396
$86$	0	0
$87$	1.58248	0.169660
$88$	0	0
$89$	7.54127	0.799373	0.399687	−	0.916652i	$-0.369119\pi$
$89$	7.54127	0.799373	0.399687	+	0.916652i	$0.369119\pi$
$90$	0	0
$91$	−3.72313	−0.390290
$92$	0	0
$93$	−2.13513	−0.221402
$94$	0	0
$95$	5.45386	0.559555
$96$	0	0
$97$	15.3021	1.55369	0.776847	−	0.629689i	$-0.216818\pi$
$97$	15.3021	1.55369	0.776847	+	0.629689i	$0.216818\pi$
$98$	0	0
$99$	4.92380	0.494861
$100$	0	0
$101$	−2.99072	−0.297588	−0.148794	−	0.988868i	$-0.547539\pi$
$101$	−2.99072	−0.297588	−0.148794	+	0.988868i	$0.547539\pi$
$102$	0	0
$103$	6.09205	0.600268	0.300134	−	0.953897i	$-0.402969\pi$
$103$	6.09205	0.600268	0.300134	+	0.953897i	$0.402969\pi$
$104$	0	0
$105$	1.21916	0.118977
$106$	0	0
$107$	12.2446	1.18373	0.591865	−	0.806037i	$-0.298392\pi$
$107$	12.2446	1.18373	0.591865	+	0.806037i	$0.298392\pi$
$108$	0	0
$109$	17.7801	1.70303	0.851514	−	0.524332i	$-0.175685\pi$
$109$	17.7801	1.70303	0.851514	+	0.524332i	$0.175685\pi$
$110$	0	0
$111$	−8.13571	−0.772207
$112$	0	0
$113$	14.4321	1.35766	0.678829	−	0.734296i	$-0.262488\pi$
$113$	14.4321	1.35766	0.678829	+	0.734296i	$0.262488\pi$
$114$	0	0
$115$	15.3955	1.43564
$116$	0	0
$117$	−5.15606	−0.476678
$118$	0	0
$119$	3.25063	0.297985
$120$	0	0
$121$	13.2439	1.20399
$122$	0	0
$123$	5.37711	0.484838
$124$	0	0
$125$	−12.0709	−1.07965
$126$	0	0
$127$	−12.6883	−1.12591	−0.562954	−	0.826488i	$-0.690335\pi$
$127$	−12.6883	−1.12591	−0.562954	+	0.826488i	$0.690335\pi$
$128$	0	0
$129$	−4.74675	−0.417928
$130$	0	0
$131$	−13.7297	−1.19957	−0.599786	−	0.800160i	$-0.704748\pi$
$131$	−13.7297	−1.19957	−0.599786	+	0.800160i	$0.704748\pi$
$132$	0	0
$133$	2.33252	0.202256
$134$	0	0
$135$	1.68837	0.145312
$136$	0	0
$137$	−15.1614	−1.29533	−0.647664	−	0.761926i	$-0.724254\pi$
$137$	−15.1614	−1.29533	−0.647664	+	0.761926i	$0.724254\pi$
$138$	0	0
$139$	−9.47213	−0.803415	−0.401708	−	0.915768i	$-0.631583\pi$
$139$	−9.47213	−0.803415	−0.401708	+	0.915768i	$0.631583\pi$
$140$	0	0
$141$	−3.74876	−0.315702
$142$	0	0
$143$	−25.3874	−2.12300
$144$	0	0
$145$	2.67182	0.221883
$146$	0	0
$147$	−6.47859	−0.534345
$148$	0	0
$149$	15.3592	1.25828	0.629139	−	0.777293i	$-0.283408\pi$
$149$	15.3592	1.25828	0.629139	+	0.777293i	$0.283408\pi$
$150$	0	0
$151$	−12.8576	−1.04634	−0.523170	−	0.852228i	$-0.675251\pi$
$151$	−12.8576	−1.04634	−0.523170	+	0.852228i	$0.675251\pi$
$152$	0	0
$153$	4.50170	0.363941
$154$	0	0
$155$	−3.60489	−0.289552
$156$	0	0
$157$	−4.41216	−0.352129	−0.176064	−	0.984379i	$-0.556337\pi$
$157$	−4.41216	−0.352129	−0.176064	+	0.984379i	$0.556337\pi$
$158$	0	0
$159$	1.47444	0.116931
$160$	0	0
$161$	6.58439	0.518923
$162$	0	0
$163$	−8.45760	−0.662450	−0.331225	−	0.943552i	$-0.607462\pi$
$163$	−8.45760	−0.662450	−0.331225	+	0.943552i	$0.607462\pi$
$164$	0	0
$165$	8.31323	0.647184
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	13.5849	1.04499
$170$	0	0
$171$	3.23024	0.247023
$172$	0	0
$173$	−4.24683	−0.322880	−0.161440	−	0.986882i	$-0.551614\pi$
$173$	−4.24683	−0.322880	−0.161440	+	0.986882i	$0.551614\pi$
$174$	0	0
$175$	−1.55205	−0.117324
$176$	0	0
$177$	3.12417	0.234827
$178$	0	0
$179$	0.599313	0.0447948	0.0223974	−	0.999749i	$-0.492870\pi$
$179$	0.599313	0.0447948	0.0223974	+	0.999749i	$0.492870\pi$
$180$	0	0
$181$	−24.3349	−1.80880	−0.904400	−	0.426686i	$-0.859681\pi$
$181$	−24.3349	−1.80880	−0.904400	+	0.426686i	$0.859681\pi$
$182$	0	0
$183$	9.12058	0.674213
$184$	0	0
$185$	−13.7361	−1.00990
$186$	0	0
$187$	22.1655	1.62090
$188$	0	0
$189$	0.722089	0.0525242
$190$	0	0
$191$	13.3821	0.968294	0.484147	−	0.874987i	$-0.339130\pi$
$191$	13.3821	0.968294	0.484147	+	0.874987i	$0.339130\pi$
$192$	0	0
$193$	−11.8898	−0.855847	−0.427924	−	0.903815i	$-0.640755\pi$
$193$	−11.8898	−0.855847	−0.427924	+	0.903815i	$0.640755\pi$
$194$	0	0
$195$	−8.70535	−0.623403
$196$	0	0
$197$	18.9713	1.35165	0.675823	−	0.737064i	$-0.263788\pi$
$197$	18.9713	1.35165	0.675823	+	0.737064i	$0.263788\pi$
$198$	0	0
$199$	20.3938	1.44568	0.722838	−	0.691017i	$-0.242837\pi$
$199$	20.3938	1.44568	0.722838	+	0.691017i	$0.242837\pi$
$200$	0	0
$201$	11.6505	0.821762
$202$	0	0
$203$	1.14269	0.0802013
$204$	0	0
$205$	9.07857	0.634075
$206$	0	0
$207$	9.11854	0.633782
$208$	0	0
$209$	15.9051	1.10018
$210$	0	0
$211$	2.39036	0.164559	0.0822797	−	0.996609i	$-0.473780\pi$
$211$	2.39036	0.164559	0.0822797	+	0.996609i	$0.473780\pi$
$212$	0	0
$213$	−13.6974	−0.938533
$214$	0	0
$215$	−8.01429	−0.546570
$216$	0	0
$217$	−1.54175	−0.104661
$218$	0	0
$219$	−2.51026	−0.169628
$220$	0	0
$221$	−23.2110	−1.56134
$222$	0	0
$223$	16.4864	1.10401	0.552006	−	0.833840i	$-0.313863\pi$
$223$	16.4864	1.10401	0.552006	+	0.833840i	$0.313863\pi$
$224$	0	0
$225$	−2.14939	−0.143293
$226$	0	0
$227$	5.37540	0.356778	0.178389	−	0.983960i	$-0.442911\pi$
$227$	5.37540	0.356778	0.178389	+	0.983960i	$0.442911\pi$
$228$	0	0
$229$	−7.85562	−0.519114	−0.259557	−	0.965728i	$-0.583577\pi$
$229$	−7.85562	−0.519114	−0.259557	+	0.965728i	$0.583577\pi$
$230$	0	0
$231$	3.55542	0.233930
$232$	0	0
$233$	5.08638	0.333220	0.166610	−	0.986023i	$-0.446718\pi$
$233$	5.08638	0.333220	0.166610	+	0.986023i	$0.446718\pi$
$234$	0	0
$235$	−6.32930	−0.412878
$236$	0	0
$237$	−12.4674	−0.809842
$238$	0	0
$239$	−12.8450	−0.830877	−0.415438	−	0.909621i	$-0.636372\pi$
$239$	−12.8450	−0.830877	−0.415438	+	0.909621i	$0.636372\pi$
$240$	0	0
$241$	5.56580	0.358524	0.179262	−	0.983801i	$-0.442629\pi$
$241$	5.56580	0.358524	0.179262	+	0.983801i	$0.442629\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.9383	−0.698821
$246$	0	0
$247$	−16.6553	−1.05975
$248$	0	0
$249$	−2.96218	−0.187720
$250$	0	0
$251$	−13.4878	−0.851343	−0.425671	−	0.904878i	$-0.639962\pi$
$251$	−13.4878	−0.851343	−0.425671	+	0.904878i	$0.639962\pi$
$252$	0	0
$253$	44.8979	2.82271
$254$	0	0
$255$	7.60056	0.475965
$256$	0	0
$257$	15.0902	0.941299	0.470650	−	0.882320i	$-0.344020\pi$
$257$	15.0902	0.941299	0.470650	+	0.882320i	$0.344020\pi$
$258$	0	0
$259$	−5.87470	−0.365036
$260$	0	0
$261$	1.58248	0.0979531
$262$	0	0
$263$	11.8686	0.731847	0.365924	−	0.930645i	$-0.380753\pi$
$263$	11.8686	0.731847	0.365924	+	0.930645i	$0.380753\pi$
$264$	0	0
$265$	2.48941	0.152923
$266$	0	0
$267$	7.54127	0.461518
$268$	0	0
$269$	0.857197	0.0522642	0.0261321	−	0.999658i	$-0.491681\pi$
$269$	0.857197	0.0522642	0.0261321	+	0.999658i	$0.491681\pi$
$270$	0	0
$271$	16.9528	1.02981	0.514905	−	0.857247i	$-0.327827\pi$
$271$	16.9528	1.02981	0.514905	+	0.857247i	$0.327827\pi$
$272$	0	0
$273$	−3.72313	−0.225334
$274$	0	0
$275$	−10.5832	−0.638190
$276$	0	0
$277$	31.3840	1.88568	0.942840	−	0.333247i	$-0.108144\pi$
$277$	31.3840	1.88568	0.942840	+	0.333247i	$0.108144\pi$
$278$	0	0
$279$	−2.13513	−0.127827
$280$	0	0
$281$	−8.26822	−0.493241	−0.246620	−	0.969112i	$-0.579320\pi$
$281$	−8.26822	−0.493241	−0.246620	+	0.969112i	$0.579320\pi$
$282$	0	0
$283$	−6.24839	−0.371428	−0.185714	−	0.982604i	$-0.559460\pi$
$283$	−6.24839	−0.371428	−0.185714	+	0.982604i	$0.559460\pi$
$284$	0	0
$285$	5.45386	0.323059
$286$	0	0
$287$	3.88275	0.229192
$288$	0	0
$289$	3.26531	0.192077
$290$	0	0
$291$	15.3021	0.897026
$292$	0	0
$293$	−22.7074	−1.32658	−0.663289	−	0.748363i	$-0.730840\pi$
$293$	−22.7074	−1.32658	−0.663289	+	0.748363i	$0.730840\pi$
$294$	0	0
$295$	5.27478	0.307109
$296$	0	0
$297$	4.92380	0.285708
$298$	0	0
$299$	−47.0157	−2.71899
$300$	0	0
$301$	−3.42757	−0.197562
$302$	0	0
$303$	−2.99072	−0.171813
$304$	0	0
$305$	15.3990	0.881742
$306$	0	0
$307$	−9.71355	−0.554381	−0.277191	−	0.960815i	$-0.589403\pi$
$307$	−9.71355	−0.554381	−0.277191	+	0.960815i	$0.589403\pi$
$308$	0	0
$309$	6.09205	0.346565
$310$	0	0
$311$	−30.6568	−1.73839	−0.869193	−	0.494474i	$-0.835361\pi$
$311$	−30.6568	−1.73839	−0.869193	+	0.494474i	$0.835361\pi$
$312$	0	0
$313$	3.88725	0.219720	0.109860	−	0.993947i	$-0.464960\pi$
$313$	3.88725	0.219720	0.109860	+	0.993947i	$0.464960\pi$
$314$	0	0
$315$	1.21916	0.0686917
$316$	0	0
$317$	−18.2051	−1.02250	−0.511250	−	0.859432i	$-0.670817\pi$
$317$	−18.2051	−1.02250	−0.511250	+	0.859432i	$0.670817\pi$
$318$	0	0
$319$	7.79183	0.436259
$320$	0	0
$321$	12.2446	0.683427
$322$	0	0
$323$	14.5416	0.809116
$324$	0	0
$325$	11.0824	0.614740
$326$	0	0
$327$	17.7801	0.983243
$328$	0	0
$329$	−2.70694	−0.149238
$330$	0	0
$331$	24.8334	1.36497	0.682483	−	0.730902i	$-0.260900\pi$
$331$	24.8334	1.36497	0.682483	+	0.730902i	$0.260900\pi$
$332$	0	0
$333$	−8.13571	−0.445834
$334$	0	0
$335$	19.6704	1.07471
$336$	0	0
$337$	−1.67704	−0.0913543	−0.0456771	−	0.998956i	$-0.514545\pi$
$337$	−1.67704	−0.0913543	−0.0456771	+	0.998956i	$0.514545\pi$
$338$	0	0
$339$	14.4321	0.783845
$340$	0	0
$341$	−10.5129	−0.569308
$342$	0	0
$343$	−9.73274	−0.525519
$344$	0	0
$345$	15.3955	0.828866
$346$	0	0
$347$	13.4895	0.724152	0.362076	−	0.932149i	$-0.382068\pi$
$347$	13.4895	0.724152	0.362076	+	0.932149i	$0.382068\pi$
$348$	0	0
$349$	6.01732	0.322099	0.161050	−	0.986946i	$-0.448512\pi$
$349$	6.01732	0.322099	0.161050	+	0.986946i	$0.448512\pi$
$350$	0	0
$351$	−5.15606	−0.275210
$352$	0	0
$353$	22.0017	1.17103	0.585515	−	0.810661i	$-0.300892\pi$
$353$	22.0017	1.17103	0.585515	+	0.810661i	$0.300892\pi$
$354$	0	0
$355$	−23.1264	−1.22742
$356$	0	0
$357$	3.25063	0.172041
$358$	0	0
$359$	20.7777	1.09661	0.548303	−	0.836280i	$-0.315274\pi$
$359$	20.7777	1.09661	0.548303	+	0.836280i	$0.315274\pi$
$360$	0	0
$361$	−8.56552	−0.450817
$362$	0	0
$363$	13.2439	0.695122
$364$	0	0
$365$	−4.23826	−0.221841
$366$	0	0
$367$	−30.9260	−1.61432	−0.807162	−	0.590330i	$-0.798998\pi$
$367$	−30.9260	−1.61432	−0.807162	+	0.590330i	$0.798998\pi$
$368$	0	0
$369$	5.37711	0.279921
$370$	0	0
$371$	1.06468	0.0552753
$372$	0	0
$373$	−1.24001	−0.0642053	−0.0321027	−	0.999485i	$-0.510220\pi$
$373$	−1.24001	−0.0642053	−0.0321027	+	0.999485i	$0.510220\pi$
$374$	0	0
$375$	−12.0709	−0.623336
$376$	0	0
$377$	−8.15936	−0.420229
$378$	0	0
$379$	−36.1456	−1.85667	−0.928337	−	0.371739i	$-0.878762\pi$
$379$	−36.1456	−1.85667	−0.928337	+	0.371739i	$0.878762\pi$
$380$	0	0
$381$	−12.6883	−0.650044
$382$	0	0
$383$	14.4479	0.738253	0.369127	−	0.929379i	$-0.379657\pi$
$383$	14.4479	0.738253	0.369127	+	0.929379i	$0.379657\pi$
$384$	0	0
$385$	6.00289	0.305936
$386$	0	0
$387$	−4.74675	−0.241291
$388$	0	0
$389$	−8.58955	−0.435508	−0.217754	−	0.976004i	$-0.569873\pi$
$389$	−8.58955	−0.435508	−0.217754	+	0.976004i	$0.569873\pi$
$390$	0	0
$391$	41.0489	2.07593
$392$	0	0
$393$	−13.7297	−0.692573
$394$	0	0
$395$	−21.0496	−1.05912
$396$	0	0
$397$	27.3990	1.37512	0.687558	−	0.726129i	$-0.258683\pi$
$397$	27.3990	1.37512	0.687558	+	0.726129i	$0.258683\pi$
$398$	0	0
$399$	2.33252	0.116772
$400$	0	0
$401$	3.71562	0.185549	0.0927747	−	0.995687i	$-0.470426\pi$
$401$	3.71562	0.185549	0.0927747	+	0.995687i	$0.470426\pi$
$402$	0	0
$403$	11.0088	0.548389
$404$	0	0
$405$	1.68837	0.0838960
$406$	0	0
$407$	−40.0586	−1.98563
$408$	0	0
$409$	−31.4154	−1.55339	−0.776695	−	0.629877i	$-0.783105\pi$
$409$	−31.4154	−1.55339	−0.776695	+	0.629877i	$0.783105\pi$
$410$	0	0
$411$	−15.1614	−0.747858
$412$	0	0
$413$	2.25593	0.111007
$414$	0	0
$415$	−5.00126	−0.245502
$416$	0	0
$417$	−9.47213	−0.463852
$418$	0	0
$419$	30.3030	1.48040	0.740200	−	0.672387i	$-0.234731\pi$
$419$	30.3030	1.48040	0.740200	+	0.672387i	$0.234731\pi$
$420$	0	0
$421$	−31.7995	−1.54981	−0.774907	−	0.632075i	$-0.782203\pi$
$421$	−31.7995	−1.54981	−0.774907	+	0.632075i	$0.782203\pi$
$422$	0	0
$423$	−3.74876	−0.182271
$424$	0	0
$425$	−9.67592	−0.469351
$426$	0	0
$427$	6.58587	0.318713
$428$	0	0
$429$	−25.3874	−1.22572
$430$	0	0
$431$	−19.1018	−0.920101	−0.460051	−	0.887893i	$-0.652169\pi$
$431$	−19.1018	−0.920101	−0.460051	+	0.887893i	$0.652169\pi$
$432$	0	0
$433$	−2.19475	−0.105473	−0.0527365	−	0.998608i	$-0.516794\pi$
$433$	−2.19475	−0.105473	−0.0527365	+	0.998608i	$0.516794\pi$
$434$	0	0
$435$	2.67182	0.128104
$436$	0	0
$437$	29.4551	1.40903
$438$	0	0
$439$	−23.8050	−1.13615	−0.568074	−	0.822977i	$-0.692311\pi$
$439$	−23.8050	−1.13615	−0.568074	+	0.822977i	$0.692311\pi$
$440$	0	0
$441$	−6.47859	−0.308504
$442$	0	0
$443$	24.1042	1.14522	0.572612	−	0.819826i	$-0.305930\pi$
$443$	24.1042	1.14522	0.572612	+	0.819826i	$0.305930\pi$
$444$	0	0
$445$	12.7325	0.603578
$446$	0	0
$447$	15.3592	0.726467
$448$	0	0
$449$	−5.66111	−0.267164	−0.133582	−	0.991038i	$-0.542648\pi$
$449$	−5.66111	−0.267164	−0.133582	+	0.991038i	$0.542648\pi$
$450$	0	0
$451$	26.4758	1.24670
$452$	0	0
$453$	−12.8576	−0.604105
$454$	0	0
$455$	−6.28604	−0.294694
$456$	0	0
$457$	15.1307	0.707786	0.353893	−	0.935286i	$-0.384858\pi$
$457$	15.1307	0.707786	0.353893	+	0.935286i	$0.384858\pi$
$458$	0	0
$459$	4.50170	0.210121
$460$	0	0
$461$	−3.92069	−0.182605	−0.0913023	−	0.995823i	$-0.529103\pi$
$461$	−3.92069	−0.182605	−0.0913023	+	0.995823i	$0.529103\pi$
$462$	0	0
$463$	−32.3205	−1.50206	−0.751030	−	0.660268i	$-0.770443\pi$
$463$	−32.3205	−1.50206	−0.751030	+	0.660268i	$0.770443\pi$
$464$	0	0
$465$	−3.60489	−0.167173
$466$	0	0
$467$	27.8059	1.28670	0.643352	−	0.765570i	$-0.277543\pi$
$467$	27.8059	1.28670	0.643352	+	0.765570i	$0.277543\pi$
$468$	0	0
$469$	8.41269	0.388462
$470$	0	0
$471$	−4.41216	−0.203302
$472$	0	0
$473$	−23.3721	−1.07465
$474$	0	0
$475$	−6.94306	−0.318570
$476$	0	0
$477$	1.47444	0.0675100
$478$	0	0
$479$	−1.80360	−0.0824086	−0.0412043	−	0.999151i	$-0.513119\pi$
$479$	−1.80360	−0.0824086	−0.0412043	+	0.999151i	$0.513119\pi$
$480$	0	0
$481$	41.9482	1.91267
$482$	0	0
$483$	6.58439	0.299600
$484$	0	0
$485$	25.8357	1.17314
$486$	0	0
$487$	−37.5627	−1.70213	−0.851064	−	0.525062i	$-0.824042\pi$
$487$	−37.5627	−1.70213	−0.851064	+	0.525062i	$0.824042\pi$
$488$	0	0
$489$	−8.45760	−0.382466
$490$	0	0
$491$	−19.5565	−0.882570	−0.441285	−	0.897367i	$-0.645477\pi$
$491$	−19.5565	−0.882570	−0.441285	+	0.897367i	$0.645477\pi$
$492$	0	0
$493$	7.12386	0.320842
$494$	0	0
$495$	8.31323	0.373652
$496$	0	0
$497$	−9.89077	−0.443661
$498$	0	0
$499$	−10.2225	−0.457623	−0.228812	−	0.973471i	$-0.573484\pi$
$499$	−10.2225	−0.457623	−0.228812	+	0.973471i	$0.573484\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	11.2594	0.502034	0.251017	−	0.967983i	$-0.419235\pi$
$503$	11.2594	0.502034	0.251017	+	0.967983i	$0.419235\pi$
$504$	0	0
$505$	−5.04946	−0.224698
$506$	0	0
$507$	13.5849	0.603328
$508$	0	0
$509$	−7.84518	−0.347731	−0.173866	−	0.984769i	$-0.555626\pi$
$509$	−7.84518	−0.347731	−0.173866	+	0.984769i	$0.555626\pi$
$510$	0	0
$511$	−1.81263	−0.0801861
$512$	0	0
$513$	3.23024	0.142619
$514$	0	0
$515$	10.2857	0.453241
$516$	0	0
$517$	−18.4581	−0.811788
$518$	0	0
$519$	−4.24683	−0.186415
$520$	0	0
$521$	−28.2656	−1.23834	−0.619168	−	0.785258i	$-0.712530\pi$
$521$	−28.2656	−1.23834	−0.619168	+	0.785258i	$0.712530\pi$
$522$	0	0
$523$	3.43781	0.150325	0.0751624	−	0.997171i	$-0.476052\pi$
$523$	3.43781	0.150325	0.0751624	+	0.997171i	$0.476052\pi$
$524$	0	0
$525$	−1.55205	−0.0677371
$526$	0	0
$527$	−9.61170	−0.418692
$528$	0	0
$529$	60.1477	2.61512
$530$	0	0
$531$	3.12417	0.135578
$532$	0	0
$533$	−27.7247	−1.20089
$534$	0	0
$535$	20.6735	0.893792
$536$	0	0
$537$	0.599313	0.0258623
$538$	0	0
$539$	−31.8993	−1.37400
$540$	0	0
$541$	23.8036	1.02340	0.511699	−	0.859165i	$-0.329016\pi$
$541$	23.8036	1.02340	0.511699	+	0.859165i	$0.329016\pi$
$542$	0	0
$543$	−24.3349	−1.04431
$544$	0	0
$545$	30.0195	1.28589
$546$	0	0
$547$	−12.5205	−0.535337	−0.267669	−	0.963511i	$-0.586253\pi$
$547$	−12.5205	−0.535337	−0.267669	+	0.963511i	$0.586253\pi$
$548$	0	0
$549$	9.12058	0.389257
$550$	0	0
$551$	5.11180	0.217770
$552$	0	0
$553$	−9.00255	−0.382827
$554$	0	0
$555$	−13.7361	−0.583066
$556$	0	0
$557$	29.1939	1.23699	0.618493	−	0.785790i	$-0.287743\pi$
$557$	29.1939	1.23699	0.618493	+	0.785790i	$0.287743\pi$
$558$	0	0
$559$	24.4745	1.03516
$560$	0	0
$561$	22.1655	0.935828
$562$	0	0
$563$	−38.1606	−1.60828	−0.804139	−	0.594441i	$-0.797373\pi$
$563$	−38.1606	−1.60828	−0.804139	+	0.594441i	$0.797373\pi$
$564$	0	0
$565$	24.3668	1.02512
$566$	0	0
$567$	0.722089	0.0303249
$568$	0	0
$569$	20.6265	0.864709	0.432354	−	0.901704i	$-0.357683\pi$
$569$	20.6265	0.864709	0.432354	+	0.901704i	$0.357683\pi$
$570$	0	0
$571$	−24.5539	−1.02755	−0.513774	−	0.857925i	$-0.671753\pi$
$571$	−24.5539	−1.02755	−0.513774	+	0.857925i	$0.671753\pi$
$572$	0	0
$573$	13.3821	0.559045
$574$	0	0
$575$	−19.5993	−0.817348
$576$	0	0
$577$	−36.6540	−1.52593	−0.762963	−	0.646443i	$-0.776256\pi$
$577$	−36.6540	−1.52593	−0.762963	+	0.646443i	$0.776256\pi$
$578$	0	0
$579$	−11.8898	−0.494124
$580$	0	0
$581$	−2.13896	−0.0887388
$582$	0	0
$583$	7.25986	0.300673
$584$	0	0
$585$	−8.70535	−0.359922
$586$	0	0
$587$	−12.0092	−0.495671	−0.247836	−	0.968802i	$-0.579719\pi$
$587$	−12.0092	−0.495671	−0.247836	+	0.968802i	$0.579719\pi$
$588$	0	0
$589$	−6.89698	−0.284185
$590$	0	0
$591$	18.9713	0.780374
$592$	0	0
$593$	−19.6797	−0.808150	−0.404075	−	0.914726i	$-0.632407\pi$
$593$	−19.6797	−0.808150	−0.404075	+	0.914726i	$0.632407\pi$
$594$	0	0
$595$	5.48828	0.224997
$596$	0	0
$597$	20.3938	0.834662
$598$	0	0
$599$	−6.13982	−0.250866	−0.125433	−	0.992102i	$-0.540032\pi$
$599$	−6.13982	−0.250866	−0.125433	+	0.992102i	$0.540032\pi$
$600$	0	0
$601$	−28.1181	−1.14696	−0.573482	−	0.819218i	$-0.694408\pi$
$601$	−28.1181	−1.14696	−0.573482	+	0.819218i	$0.694408\pi$
$602$	0	0
$603$	11.6505	0.474445
$604$	0	0
$605$	22.3606	0.909087
$606$	0	0
$607$	13.8152	0.560743	0.280372	−	0.959892i	$-0.409542\pi$
$607$	13.8152	0.560743	0.280372	+	0.959892i	$0.409542\pi$
$608$	0	0
$609$	1.14269	0.0463042
$610$	0	0
$611$	19.3288	0.781960
$612$	0	0
$613$	17.5891	0.710416	0.355208	−	0.934787i	$-0.384410\pi$
$613$	17.5891	0.710416	0.355208	+	0.934787i	$0.384410\pi$
$614$	0	0
$615$	9.07857	0.366083
$616$	0	0
$617$	−7.71765	−0.310701	−0.155351	−	0.987859i	$-0.549651\pi$
$617$	−7.71765	−0.310701	−0.155351	+	0.987859i	$0.549651\pi$
$618$	0	0
$619$	25.1563	1.01112	0.505559	−	0.862792i	$-0.331286\pi$
$619$	25.1563	1.01112	0.505559	+	0.862792i	$0.331286\pi$
$620$	0	0
$621$	9.11854	0.365914
$622$	0	0
$623$	5.44547	0.218168
$624$	0	0
$625$	−9.63315	−0.385326
$626$	0	0
$627$	15.9051	0.635188
$628$	0	0
$629$	−36.6245	−1.46032
$630$	0	0
$631$	−32.0287	−1.27504	−0.637521	−	0.770433i	$-0.720040\pi$
$631$	−32.0287	−1.27504	−0.637521	+	0.770433i	$0.720040\pi$
$632$	0	0
$633$	2.39036	0.0950084
$634$	0	0
$635$	−21.4227	−0.850133
$636$	0	0
$637$	33.4040	1.32351
$638$	0	0
$639$	−13.6974	−0.541862
$640$	0	0
$641$	35.3785	1.39737	0.698683	−	0.715431i	$-0.253770\pi$
$641$	35.3785	1.39737	0.698683	+	0.715431i	$0.253770\pi$
$642$	0	0
$643$	−8.42726	−0.332339	−0.166169	−	0.986097i	$-0.553140\pi$
$643$	−8.42726	−0.332339	−0.166169	+	0.986097i	$0.553140\pi$
$644$	0	0
$645$	−8.01429	−0.315562
$646$	0	0
$647$	45.0291	1.77028	0.885138	−	0.465328i	$-0.154064\pi$
$647$	45.0291	1.77028	0.885138	+	0.465328i	$0.154064\pi$
$648$	0	0
$649$	15.3828	0.603829
$650$	0	0
$651$	−1.54175	−0.0604260
$652$	0	0
$653$	−44.0405	−1.72344	−0.861719	−	0.507386i	$-0.830612\pi$
$653$	−44.0405	−1.72344	−0.861719	+	0.507386i	$0.830612\pi$
$654$	0	0
$655$	−23.1809	−0.905754
$656$	0	0
$657$	−2.51026	−0.0979346
$658$	0	0
$659$	−49.6642	−1.93464	−0.967321	−	0.253555i	$-0.918400\pi$
$659$	−49.6642	−1.93464	−0.967321	+	0.253555i	$0.918400\pi$
$660$	0	0
$661$	12.2095	0.474893	0.237446	−	0.971401i	$-0.423690\pi$
$661$	12.2095	0.474893	0.237446	+	0.971401i	$0.423690\pi$
$662$	0	0
$663$	−23.2110	−0.901442
$664$	0	0
$665$	3.93817	0.152716
$666$	0	0
$667$	14.4299	0.558728
$668$	0	0
$669$	16.4864	0.637401
$670$	0	0
$671$	44.9080	1.73365
$672$	0	0
$673$	30.6693	1.18221	0.591107	−	0.806593i	$-0.298691\pi$
$673$	30.6693	1.18221	0.591107	+	0.806593i	$0.298691\pi$
$674$	0	0
$675$	−2.14939	−0.0827301
$676$	0	0
$677$	15.8194	0.607988	0.303994	−	0.952674i	$-0.401680\pi$
$677$	15.8194	0.607988	0.303994	+	0.952674i	$0.401680\pi$
$678$	0	0
$679$	11.0495	0.424041
$680$	0	0
$681$	5.37540	0.205986
$682$	0	0
$683$	−8.58263	−0.328405	−0.164203	−	0.986427i	$-0.552505\pi$
$683$	−8.58263	−0.328405	−0.164203	+	0.986427i	$0.552505\pi$
$684$	0	0
$685$	−25.5982	−0.978055
$686$	0	0
$687$	−7.85562	−0.299711
$688$	0	0
$689$	−7.60230	−0.289625
$690$	0	0
$691$	−6.22154	−0.236679	−0.118339	−	0.992973i	$-0.537757\pi$
$691$	−6.22154	−0.236679	−0.118339	+	0.992973i	$0.537757\pi$
$692$	0	0
$693$	3.55542	0.135059
$694$	0	0
$695$	−15.9925	−0.606630
$696$	0	0
$697$	24.2061	0.916873
$698$	0	0
$699$	5.08638	0.192385
$700$	0	0
$701$	−17.5201	−0.661724	−0.330862	−	0.943679i	$-0.607339\pi$
$701$	−17.5201	−0.661724	−0.330862	+	0.943679i	$0.607339\pi$
$702$	0	0
$703$	−26.2803	−0.991181
$704$	0	0
$705$	−6.32930	−0.238375
$706$	0	0
$707$	−2.15957	−0.0812190
$708$	0	0
$709$	38.8631	1.45953	0.729766	−	0.683697i	$-0.239629\pi$
$709$	38.8631	1.45953	0.729766	+	0.683697i	$0.239629\pi$
$710$	0	0
$711$	−12.4674	−0.467563
$712$	0	0
$713$	−19.4692	−0.729129
$714$	0	0
$715$	−42.8635	−1.60300
$716$	0	0
$717$	−12.8450	−0.479707
$718$	0	0
$719$	−10.3606	−0.386383	−0.193192	−	0.981161i	$-0.561884\pi$
$719$	−10.3606	−0.386383	−0.193192	+	0.981161i	$0.561884\pi$
$720$	0	0
$721$	4.39901	0.163828
$722$	0	0
$723$	5.56580	0.206994
$724$	0	0
$725$	−3.40137	−0.126324
$726$	0	0
$727$	−51.7807	−1.92044	−0.960220	−	0.279244i	$-0.909916\pi$
$727$	−51.7807	−1.92044	−0.960220	+	0.279244i	$0.909916\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.3684	−0.790340
$732$	0	0
$733$	10.7091	0.395550	0.197775	−	0.980247i	$-0.436628\pi$
$733$	10.7091	0.395550	0.197775	+	0.980247i	$0.436628\pi$
$734$	0	0
$735$	−10.9383	−0.403465
$736$	0	0
$737$	57.3647	2.11306
$738$	0	0
$739$	−40.3746	−1.48520	−0.742602	−	0.669734i	$-0.766408\pi$
$739$	−40.3746	−1.48520	−0.742602	+	0.669734i	$0.766408\pi$
$740$	0	0
$741$	−16.6553	−0.611849
$742$	0	0
$743$	−16.6651	−0.611383	−0.305692	−	0.952131i	$-0.598888\pi$
$743$	−16.6651	−0.611383	−0.305692	+	0.952131i	$0.598888\pi$
$744$	0	0
$745$	25.9321	0.950080
$746$	0	0
$747$	−2.96218	−0.108380
$748$	0	0
$749$	8.84169	0.323068
$750$	0	0
$751$	25.7654	0.940191	0.470096	−	0.882615i	$-0.344219\pi$
$751$	25.7654	0.940191	0.470096	+	0.882615i	$0.344219\pi$
$752$	0	0
$753$	−13.4878	−0.491523
$754$	0	0
$755$	−21.7085	−0.790054
$756$	0	0
$757$	−3.58635	−0.130348	−0.0651740	−	0.997874i	$-0.520760\pi$
$757$	−3.58635	−0.130348	−0.0651740	+	0.997874i	$0.520760\pi$
$758$	0	0
$759$	44.8979	1.62969
$760$	0	0
$761$	2.56070	0.0928255	0.0464127	−	0.998922i	$-0.485221\pi$
$761$	2.56070	0.0928255	0.0464127	+	0.998922i	$0.485221\pi$
$762$	0	0
$763$	12.8388	0.464797
$764$	0	0
$765$	7.60056	0.274799
$766$	0	0
$767$	−16.1084	−0.581641
$768$	0	0
$769$	−14.5797	−0.525756	−0.262878	−	0.964829i	$-0.584672\pi$
$769$	−14.5797	−0.525756	−0.262878	+	0.964829i	$0.584672\pi$
$770$	0	0
$771$	15.0902	0.543459
$772$	0	0
$773$	−21.4680	−0.772151	−0.386076	−	0.922467i	$-0.626170\pi$
$773$	−21.4680	−0.772151	−0.386076	+	0.922467i	$0.626170\pi$
$774$	0	0
$775$	4.58923	0.164850
$776$	0	0
$777$	−5.87470	−0.210754
$778$	0	0
$779$	17.3694	0.622323
$780$	0	0
$781$	−67.4435	−2.41332
$782$	0	0
$783$	1.58248	0.0565533
$784$	0	0
$785$	−7.44938	−0.265880
$786$	0	0
$787$	−2.87234	−0.102388	−0.0511939	−	0.998689i	$-0.516303\pi$
$787$	−2.87234	−0.102388	−0.0511939	+	0.998689i	$0.516303\pi$
$788$	0	0
$789$	11.8686	0.422532
$790$	0	0
$791$	10.4213	0.370538
$792$	0	0
$793$	−47.0263	−1.66995
$794$	0	0
$795$	2.48941	0.0882902
$796$	0	0
$797$	39.8716	1.41232	0.706162	−	0.708050i	$-0.250425\pi$
$797$	39.8716	1.41232	0.706162	+	0.708050i	$0.250425\pi$
$798$	0	0
$799$	−16.8758	−0.597022
$800$	0	0
$801$	7.54127	0.266458
$802$	0	0
$803$	−12.3600	−0.436176
$804$	0	0
$805$	11.1169	0.391820
$806$	0	0
$807$	0.857197	0.0301748
$808$	0	0
$809$	−41.6315	−1.46369	−0.731843	−	0.681473i	$-0.761339\pi$
$809$	−41.6315	−1.46369	−0.731843	+	0.681473i	$0.761339\pi$
$810$	0	0
$811$	42.3043	1.48551	0.742753	−	0.669565i	$-0.233520\pi$
$811$	42.3043	1.48551	0.742753	+	0.669565i	$0.233520\pi$
$812$	0	0
$813$	16.9528	0.594561
$814$	0	0
$815$	−14.2796	−0.500192
$816$	0	0
$817$	−15.3332	−0.536439
$818$	0	0
$819$	−3.72313	−0.130097
$820$	0	0
$821$	−44.9001	−1.56702	−0.783512	−	0.621377i	$-0.786574\pi$
$821$	−44.9001	−1.56702	−0.783512	+	0.621377i	$0.786574\pi$
$822$	0	0
$823$	−21.8823	−0.762769	−0.381385	−	0.924416i	$-0.624553\pi$
$823$	−21.8823	−0.762769	−0.381385	+	0.924416i	$0.624553\pi$
$824$	0	0
$825$	−10.5832	−0.368459
$826$	0	0
$827$	−17.5690	−0.610934	−0.305467	−	0.952203i	$-0.598813\pi$
$827$	−17.5690	−0.610934	−0.305467	+	0.952203i	$0.598813\pi$
$828$	0	0
$829$	10.8075	0.375361	0.187680	−	0.982230i	$-0.439903\pi$
$829$	10.8075	0.375361	0.187680	+	0.982230i	$0.439903\pi$
$830$	0	0
$831$	31.3840	1.08870
$832$	0	0
$833$	−29.1647	−1.01050
$834$	0	0
$835$	−1.68837	−0.0584286
$836$	0	0
$837$	−2.13513	−0.0738008
$838$	0	0
$839$	−29.0750	−1.00378	−0.501891	−	0.864931i	$-0.667362\pi$
$839$	−29.0750	−1.00378	−0.501891	+	0.864931i	$0.667362\pi$
$840$	0	0
$841$	−26.4958	−0.913647
$842$	0	0
$843$	−8.26822	−0.284773
$844$	0	0
$845$	22.9364	0.789038
$846$	0	0
$847$	9.56324	0.328597
$848$	0	0
$849$	−6.24839	−0.214444
$850$	0	0
$851$	−74.1858	−2.54305
$852$	0	0
$853$	48.6140	1.66451	0.832256	−	0.554392i	$-0.187049\pi$
$853$	48.6140	1.66451	0.832256	+	0.554392i	$0.187049\pi$
$854$	0	0
$855$	5.45386	0.186518
$856$	0	0
$857$	36.4623	1.24553	0.622763	−	0.782410i	$-0.286010\pi$
$857$	36.4623	1.24553	0.622763	+	0.782410i	$0.286010\pi$
$858$	0	0
$859$	−8.60352	−0.293548	−0.146774	−	0.989170i	$-0.546889\pi$
$859$	−8.60352	−0.293548	−0.146774	+	0.989170i	$0.546889\pi$
$860$	0	0
$861$	3.88275	0.132324
$862$	0	0
$863$	−26.4936	−0.901852	−0.450926	−	0.892561i	$-0.648906\pi$
$863$	−26.4936	−0.901852	−0.450926	+	0.892561i	$0.648906\pi$
$864$	0	0
$865$	−7.17024	−0.243795
$866$	0	0
$867$	3.26531	0.110896
$868$	0	0
$869$	−61.3869	−2.08241
$870$	0	0
$871$	−60.0706	−2.03541
$872$	0	0
$873$	15.3021	0.517898
$874$	0	0
$875$	−8.71623	−0.294662
$876$	0	0
$877$	17.8702	0.603432	0.301716	−	0.953398i	$-0.402441\pi$
$877$	17.8702	0.603432	0.301716	+	0.953398i	$0.402441\pi$
$878$	0	0
$879$	−22.7074	−0.765900
$880$	0	0
$881$	23.2404	0.782990	0.391495	−	0.920180i	$-0.371958\pi$
$881$	23.2404	0.782990	0.391495	+	0.920180i	$0.371958\pi$
$882$	0	0
$883$	33.2781	1.11990	0.559948	−	0.828528i	$-0.310821\pi$
$883$	33.2781	1.11990	0.559948	+	0.828528i	$0.310821\pi$
$884$	0	0
$885$	5.27478	0.177310
$886$	0	0
$887$	15.3015	0.513775	0.256887	−	0.966441i	$-0.417303\pi$
$887$	15.3015	0.513775	0.256887	+	0.966441i	$0.417303\pi$
$888$	0	0
$889$	−9.16212	−0.307288
$890$	0	0
$891$	4.92380	0.164954
$892$	0	0
$893$	−12.1094	−0.405226
$894$	0	0
$895$	1.01187	0.0338229
$896$	0	0
$897$	−47.0157	−1.56981
$898$	0	0
$899$	−3.37880	−0.112689
$900$	0	0
$901$	6.63749	0.221127
$902$	0	0
$903$	−3.42757	−0.114063
$904$	0	0
$905$	−41.0864	−1.36576
$906$	0	0
$907$	26.7007	0.886583	0.443292	−	0.896377i	$-0.353811\pi$
$907$	26.7007	0.886583	0.443292	+	0.896377i	$0.353811\pi$
$908$	0	0
$909$	−2.99072	−0.0991961
$910$	0	0
$911$	−15.6323	−0.517920	−0.258960	−	0.965888i	$-0.583380\pi$
$911$	−15.6323	−0.517920	−0.258960	+	0.965888i	$0.583380\pi$
$912$	0	0
$913$	−14.5852	−0.482699
$914$	0	0
$915$	15.3990	0.509074
$916$	0	0
$917$	−9.91409	−0.327392
$918$	0	0
$919$	−18.3950	−0.606795	−0.303398	−	0.952864i	$-0.598121\pi$
$919$	−18.3950	−0.606795	−0.303398	+	0.952864i	$0.598121\pi$
$920$	0	0
$921$	−9.71355	−0.320072
$922$	0	0
$923$	70.6248	2.32464
$924$	0	0
$925$	17.4868	0.574963
$926$	0	0
$927$	6.09205	0.200089
$928$	0	0
$929$	18.2625	0.599174	0.299587	−	0.954069i	$-0.403151\pi$
$929$	18.2625	0.599174	0.299587	+	0.954069i	$0.403151\pi$
$930$	0	0
$931$	−20.9274	−0.685869
$932$	0	0
$933$	−30.6568	−1.00366
$934$	0	0
$935$	37.4236	1.22388
$936$	0	0
$937$	3.89282	0.127173	0.0635864	−	0.997976i	$-0.479746\pi$
$937$	3.89282	0.127173	0.0635864	+	0.997976i	$0.479746\pi$
$938$	0	0
$939$	3.88725	0.126856
$940$	0	0
$941$	−46.3525	−1.51105	−0.755524	−	0.655121i	$-0.772618\pi$
$941$	−46.3525	−1.51105	−0.755524	+	0.655121i	$0.772618\pi$
$942$	0	0
$943$	49.0314	1.59668
$944$	0	0
$945$	1.21916	0.0396592
$946$	0	0
$947$	−10.6858	−0.347243	−0.173622	−	0.984812i	$-0.555547\pi$
$947$	−10.6858	−0.347243	−0.173622	+	0.984812i	$0.555547\pi$
$948$	0	0
$949$	12.9431	0.420149
$950$	0	0
$951$	−18.2051	−0.590340
$952$	0	0
$953$	−9.70387	−0.314339	−0.157170	−	0.987572i	$-0.550237\pi$
$953$	−9.70387	−0.314339	−0.157170	+	0.987572i	$0.550237\pi$
$954$	0	0
$955$	22.5940	0.731124
$956$	0	0
$957$	7.79183	0.251874
$958$	0	0
$959$	−10.9479	−0.353526
$960$	0	0
$961$	−26.4412	−0.852943
$962$	0	0
$963$	12.2446	0.394577
$964$	0	0
$965$	−20.0744	−0.646219
$966$	0	0
$967$	2.89854	0.0932107	0.0466054	−	0.998913i	$-0.485160\pi$
$967$	2.89854	0.0932107	0.0466054	+	0.998913i	$0.485160\pi$
$968$	0	0
$969$	14.5416	0.467143
$970$	0	0
$971$	−26.8901	−0.862944	−0.431472	−	0.902126i	$-0.642006\pi$
$971$	−26.8901	−0.862944	−0.431472	+	0.902126i	$0.642006\pi$
$972$	0	0
$973$	−6.83972	−0.219271
$974$	0	0
$975$	11.0824	0.354921
$976$	0	0
$977$	−27.0951	−0.866849	−0.433425	−	0.901190i	$-0.642695\pi$
$977$	−27.0951	−0.866849	−0.433425	+	0.901190i	$0.642695\pi$
$978$	0	0
$979$	37.1317	1.18674
$980$	0	0
$981$	17.7801	0.567676
$982$	0	0
$983$	−37.8416	−1.20696	−0.603481	−	0.797378i	$-0.706220\pi$
$983$	−37.8416	−1.20696	−0.603481	+	0.797378i	$0.706220\pi$
$984$	0	0
$985$	32.0306	1.02058
$986$	0	0
$987$	−2.70694	−0.0861627
$988$	0	0
$989$	−43.2834	−1.37633
$990$	0	0
$991$	−36.3760	−1.15552	−0.577761	−	0.816206i	$-0.696073\pi$
$991$	−36.3760	−1.15552	−0.577761	+	0.816206i	$0.696073\pi$
$992$	0	0
$993$	24.8334	0.788063
$994$	0	0
$995$	34.4323	1.09158
$996$	0	0
$997$	−59.9879	−1.89984	−0.949918	−	0.312501i	$-0.898833\pi$
$997$	−59.9879	−1.89984	−0.949918	+	0.312501i	$0.898833\pi$
$998$	0	0
$999$	−8.13571	−0.257402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.m.1.9	✓	13
4.3	odd	2		8016.2.a.bg.1.9		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.9	✓	13	1.1	even	1	trivial
8016.2.a.bg.1.9		13	4.3	odd	2

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.68837 q^{5} +0.722089 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.68837 q^{5} +0.722089 q^{7} +1.00000 q^{9} +4.92380 q^{11} -5.15606 q^{13} +1.68837 q^{15} +4.50170 q^{17} +3.23024 q^{19} +0.722089 q^{21} +9.11854 q^{23} -2.14939 q^{25} +1.00000 q^{27} +1.58248 q^{29} -2.13513 q^{31} +4.92380 q^{33} +1.21916 q^{35} -8.13571 q^{37} -5.15606 q^{39} +5.37711 q^{41} -4.74675 q^{43} +1.68837 q^{45} -3.74876 q^{47} -6.47859 q^{49} +4.50170 q^{51} +1.47444 q^{53} +8.31323 q^{55} +3.23024 q^{57} +3.12417 q^{59} +9.12058 q^{61} +0.722089 q^{63} -8.70535 q^{65} +11.6505 q^{67} +9.11854 q^{69} -13.6974 q^{71} -2.51026 q^{73} -2.14939 q^{75} +3.55542 q^{77} -12.4674 q^{79} +1.00000 q^{81} -2.96218 q^{83} +7.60056 q^{85} +1.58248 q^{87} +7.54127 q^{89} -3.72313 q^{91} -2.13513 q^{93} +5.45386 q^{95} +15.3021 q^{97} +4.92380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * q^99

Introduction

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