LMFDB - Embedded newform 4008.2.a.m.1.1

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			1.1
Root			$-4.02595$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -4.02595 q^{5} +0.910115 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.02595 q^{5} +0.910115 q^{7} +1.00000 q^{9} -5.64943 q^{11} +6.43968 q^{13} -4.02595 q^{15} +3.30990 q^{17} +1.28987 q^{19} +0.910115 q^{21} -9.24783 q^{23} +11.2083 q^{25} +1.00000 q^{27} +2.90838 q^{29} -10.6281 q^{31} -5.64943 q^{33} -3.66408 q^{35} +3.18976 q^{37} +6.43968 q^{39} +1.79168 q^{41} +2.55965 q^{43} -4.02595 q^{45} +1.91679 q^{47} -6.17169 q^{49} +3.30990 q^{51} -6.55631 q^{53} +22.7443 q^{55} +1.28987 q^{57} -2.97385 q^{59} +8.34803 q^{61} +0.910115 q^{63} -25.9258 q^{65} +4.23967 q^{67} -9.24783 q^{69} +10.1160 q^{71} +1.27643 q^{73} +11.2083 q^{75} -5.14163 q^{77} -1.79214 q^{79} +1.00000 q^{81} +13.8633 q^{83} -13.3255 q^{85} +2.90838 q^{87} +5.45522 q^{89} +5.86085 q^{91} -10.6281 q^{93} -5.19293 q^{95} +13.0108 q^{97} -5.64943 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.02595	−1.80046	−0.900229	−	0.435416i	$-0.856601\pi$
$5$	−4.02595	−1.80046	−0.900229	+	0.435416i	$0.856601\pi$
$6$	0	0
$7$	0.910115	0.343991	0.171996	−	0.985098i	$-0.444979\pi$
$7$	0.910115	0.343991	0.171996	+	0.985098i	$0.444979\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.64943	−1.70337	−0.851683	−	0.524058i	$-0.824418\pi$
$11$	−5.64943	−1.70337	−0.851683	+	0.524058i	$0.824418\pi$
$12$	0	0
$13$	6.43968	1.78605	0.893023	−	0.450011i	$-0.148580\pi$
$13$	6.43968	1.78605	0.893023	+	0.450011i	$0.148580\pi$
$14$	0	0
$15$	−4.02595	−1.03950
$16$	0	0
$17$	3.30990	0.802770	0.401385	−	0.915909i	$-0.368529\pi$
$17$	3.30990	0.802770	0.401385	+	0.915909i	$0.368529\pi$
$18$	0	0
$19$	1.28987	0.295915	0.147958	−	0.988994i	$-0.452730\pi$
$19$	1.28987	0.295915	0.147958	+	0.988994i	$0.452730\pi$
$20$	0	0
$21$	0.910115	0.198603
$22$	0	0
$23$	−9.24783	−1.92830	−0.964152	−	0.265349i	$-0.914513\pi$
$23$	−9.24783	−1.92830	−0.964152	+	0.265349i	$0.914513\pi$
$24$	0	0
$25$	11.2083	2.24165
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.90838	0.540072	0.270036	−	0.962850i	$-0.412964\pi$
$29$	2.90838	0.540072	0.270036	+	0.962850i	$0.412964\pi$
$30$	0	0
$31$	−10.6281	−1.90886	−0.954431	−	0.298432i	$-0.903536\pi$
$31$	−10.6281	−1.90886	−0.954431	+	0.298432i	$0.903536\pi$
$32$	0	0
$33$	−5.64943	−0.983439
$34$	0	0
$35$	−3.66408	−0.619342
$36$	0	0
$37$	3.18976	0.524393	0.262196	−	0.965015i	$-0.415553\pi$
$37$	3.18976	0.524393	0.262196	+	0.965015i	$0.415553\pi$
$38$	0	0
$39$	6.43968	1.03117
$40$	0	0
$41$	1.79168	0.279813	0.139907	−	0.990165i	$-0.455320\pi$
$41$	1.79168	0.279813	0.139907	+	0.990165i	$0.455320\pi$
$42$	0	0
$43$	2.55965	0.390344	0.195172	−	0.980769i	$-0.437474\pi$
$43$	2.55965	0.390344	0.195172	+	0.980769i	$0.437474\pi$
$44$	0	0
$45$	−4.02595	−0.600153
$46$	0	0
$47$	1.91679	0.279593	0.139797	−	0.990180i	$-0.455355\pi$
$47$	1.91679	0.279593	0.139797	+	0.990180i	$0.455355\pi$
$48$	0	0
$49$	−6.17169	−0.881670
$50$	0	0
$51$	3.30990	0.463479
$52$	0	0
$53$	−6.55631	−0.900579	−0.450289	−	0.892883i	$-0.648679\pi$
$53$	−6.55631	−0.900579	−0.450289	+	0.892883i	$0.648679\pi$
$54$	0	0
$55$	22.7443	3.06684
$56$	0	0
$57$	1.28987	0.170847
$58$	0	0
$59$	−2.97385	−0.387162	−0.193581	−	0.981084i	$-0.562010\pi$
$59$	−2.97385	−0.387162	−0.193581	+	0.981084i	$0.562010\pi$
$60$	0	0
$61$	8.34803	1.06886	0.534428	−	0.845214i	$-0.320527\pi$
$61$	8.34803	1.06886	0.534428	+	0.845214i	$0.320527\pi$
$62$	0	0
$63$	0.910115	0.114664
$64$	0	0
$65$	−25.9258	−3.21570
$66$	0	0
$67$	4.23967	0.517959	0.258979	−	0.965883i	$-0.416614\pi$
$67$	4.23967	0.517959	0.258979	+	0.965883i	$0.416614\pi$
$68$	0	0
$69$	−9.24783	−1.11331
$70$	0	0
$71$	10.1160	1.20055	0.600276	−	0.799793i	$-0.295057\pi$
$71$	10.1160	1.20055	0.600276	+	0.799793i	$0.295057\pi$
$72$	0	0
$73$	1.27643	0.149395	0.0746974	−	0.997206i	$-0.476201\pi$
$73$	1.27643	0.149395	0.0746974	+	0.997206i	$0.476201\pi$
$74$	0	0
$75$	11.2083	1.29422
$76$	0	0
$77$	−5.14163	−0.585943
$78$	0	0
$79$	−1.79214	−0.201631	−0.100816	−	0.994905i	$-0.532145\pi$
$79$	−1.79214	−0.201631	−0.100816	+	0.994905i	$0.532145\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8633	1.52169	0.760846	−	0.648932i	$-0.224784\pi$
$83$	13.8633	1.52169	0.760846	+	0.648932i	$0.224784\pi$
$84$	0	0
$85$	−13.3255	−1.44535
$86$	0	0
$87$	2.90838	0.311811
$88$	0	0
$89$	5.45522	0.578252	0.289126	−	0.957291i	$-0.406635\pi$
$89$	5.45522	0.578252	0.289126	+	0.957291i	$0.406635\pi$
$90$	0	0
$91$	5.86085	0.614384
$92$	0	0
$93$	−10.6281	−1.10208
$94$	0	0
$95$	−5.19293	−0.532783
$96$	0	0
$97$	13.0108	1.32105	0.660525	−	0.750804i	$-0.270334\pi$
$97$	13.0108	1.32105	0.660525	+	0.750804i	$0.270334\pi$
$98$	0	0
$99$	−5.64943	−0.567789
$100$	0	0
$101$	13.5075	1.34405	0.672023	−	0.740530i	$-0.265425\pi$
$101$	13.5075	1.34405	0.672023	+	0.740530i	$0.265425\pi$
$102$	0	0
$103$	−2.77840	−0.273763	−0.136882	−	0.990587i	$-0.543708\pi$
$103$	−2.77840	−0.273763	−0.136882	+	0.990587i	$0.543708\pi$
$104$	0	0
$105$	−3.66408	−0.357577
$106$	0	0
$107$	3.24267	0.313480	0.156740	−	0.987640i	$-0.449901\pi$
$107$	3.24267	0.313480	0.156740	+	0.987640i	$0.449901\pi$
$108$	0	0
$109$	9.99492	0.957339	0.478670	−	0.877995i	$-0.341119\pi$
$109$	9.99492	0.957339	0.478670	+	0.877995i	$0.341119\pi$
$110$	0	0
$111$	3.18976	0.302758
$112$	0	0
$113$	12.1305	1.14114	0.570571	−	0.821249i	$-0.306722\pi$
$113$	12.1305	1.14114	0.570571	+	0.821249i	$0.306722\pi$
$114$	0	0
$115$	37.2313	3.47183
$116$	0	0
$117$	6.43968	0.595349
$118$	0	0
$119$	3.01239	0.276146
$120$	0	0
$121$	20.9160	1.90145
$122$	0	0
$123$	1.79168	0.161550
$124$	0	0
$125$	−24.9941	−2.23554
$126$	0	0
$127$	12.6803	1.12520	0.562599	−	0.826730i	$-0.309802\pi$
$127$	12.6803	1.12520	0.562599	+	0.826730i	$0.309802\pi$
$128$	0	0
$129$	2.55965	0.225365
$130$	0	0
$131$	7.08979	0.619438	0.309719	−	0.950828i	$-0.399765\pi$
$131$	7.08979	0.619438	0.309719	+	0.950828i	$0.399765\pi$
$132$	0	0
$133$	1.17393	0.101792
$134$	0	0
$135$	−4.02595	−0.346498
$136$	0	0
$137$	1.44562	0.123508	0.0617538	−	0.998091i	$-0.480331\pi$
$137$	1.44562	0.123508	0.0617538	+	0.998091i	$0.480331\pi$
$138$	0	0
$139$	−19.4215	−1.64731	−0.823656	−	0.567089i	$-0.808069\pi$
$139$	−19.4215	−1.64731	−0.823656	+	0.567089i	$0.808069\pi$
$140$	0	0
$141$	1.91679	0.161423
$142$	0	0
$143$	−36.3805	−3.04229
$144$	0	0
$145$	−11.7090	−0.972378
$146$	0	0
$147$	−6.17169	−0.509032
$148$	0	0
$149$	10.6988	0.876482	0.438241	−	0.898858i	$-0.355602\pi$
$149$	10.6988	0.876482	0.438241	+	0.898858i	$0.355602\pi$
$150$	0	0
$151$	1.13414	0.0922949	0.0461474	−	0.998935i	$-0.485306\pi$
$151$	1.13414	0.0922949	0.0461474	+	0.998935i	$0.485306\pi$
$152$	0	0
$153$	3.30990	0.267590
$154$	0	0
$155$	42.7882	3.43683
$156$	0	0
$157$	−2.13747	−0.170589	−0.0852944	−	0.996356i	$-0.527183\pi$
$157$	−2.13747	−0.170589	−0.0852944	+	0.996356i	$0.527183\pi$
$158$	0	0
$159$	−6.55631	−0.519949
$160$	0	0
$161$	−8.41658	−0.663320
$162$	0	0
$163$	−8.75643	−0.685857	−0.342928	−	0.939362i	$-0.611419\pi$
$163$	−8.75643	−0.685857	−0.342928	+	0.939362i	$0.611419\pi$
$164$	0	0
$165$	22.7443	1.77064
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	28.4695	2.18996
$170$	0	0
$171$	1.28987	0.0986384
$172$	0	0
$173$	20.7771	1.57966	0.789829	−	0.613328i	$-0.210169\pi$
$173$	20.7771	1.57966	0.789829	+	0.613328i	$0.210169\pi$
$174$	0	0
$175$	10.2008	0.771108
$176$	0	0
$177$	−2.97385	−0.223528
$178$	0	0
$179$	20.6874	1.54625	0.773123	−	0.634256i	$-0.218693\pi$
$179$	20.6874	1.54625	0.773123	+	0.634256i	$0.218693\pi$
$180$	0	0
$181$	−10.6867	−0.794335	−0.397167	−	0.917746i	$-0.630007\pi$
$181$	−10.6867	−0.794335	−0.397167	+	0.917746i	$0.630007\pi$
$182$	0	0
$183$	8.34803	0.617104
$184$	0	0
$185$	−12.8418	−0.944148
$186$	0	0
$187$	−18.6991	−1.36741
$188$	0	0
$189$	0.910115	0.0662011
$190$	0	0
$191$	23.7568	1.71898	0.859489	−	0.511154i	$-0.170782\pi$
$191$	23.7568	1.71898	0.859489	+	0.511154i	$0.170782\pi$
$192$	0	0
$193$	0.301545	0.0217057	0.0108528	−	0.999941i	$-0.496545\pi$
$193$	0.301545	0.0217057	0.0108528	+	0.999941i	$0.496545\pi$
$194$	0	0
$195$	−25.9258	−1.85659
$196$	0	0
$197$	−21.9860	−1.56644	−0.783218	−	0.621747i	$-0.786423\pi$
$197$	−21.9860	−1.56644	−0.783218	+	0.621747i	$0.786423\pi$
$198$	0	0
$199$	−18.3274	−1.29920	−0.649598	−	0.760278i	$-0.725063\pi$
$199$	−18.3274	−1.29920	−0.649598	+	0.760278i	$0.725063\pi$
$200$	0	0
$201$	4.23967	0.299044
$202$	0	0
$203$	2.64696	0.185780
$204$	0	0
$205$	−7.21321	−0.503792
$206$	0	0
$207$	−9.24783	−0.642768
$208$	0	0
$209$	−7.28700	−0.504052
$210$	0	0
$211$	25.9537	1.78673	0.893363	−	0.449336i	$-0.148339\pi$
$211$	25.9537	1.78673	0.893363	+	0.449336i	$0.148339\pi$
$212$	0	0
$213$	10.1160	0.693139
$214$	0	0
$215$	−10.3050	−0.702798
$216$	0	0
$217$	−9.67279	−0.656631
$218$	0	0
$219$	1.27643	0.0862531
$220$	0	0
$221$	21.3147	1.43378
$222$	0	0
$223$	−23.8575	−1.59761	−0.798807	−	0.601588i	$-0.794535\pi$
$223$	−23.8575	−1.59761	−0.798807	+	0.601588i	$0.794535\pi$
$224$	0	0
$225$	11.2083	0.747217
$226$	0	0
$227$	11.8006	0.783233	0.391616	−	0.920129i	$-0.371916\pi$
$227$	11.8006	0.783233	0.391616	+	0.920129i	$0.371916\pi$
$228$	0	0
$229$	4.93628	0.326199	0.163099	−	0.986610i	$-0.447851\pi$
$229$	4.93628	0.326199	0.163099	+	0.986610i	$0.447851\pi$
$230$	0	0
$231$	−5.14163	−0.338294
$232$	0	0
$233$	20.2292	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$233$	20.2292	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$234$	0	0
$235$	−7.71692	−0.503396
$236$	0	0
$237$	−1.79214	−0.116412
$238$	0	0
$239$	−9.31462	−0.602513	−0.301256	−	0.953543i	$-0.597406\pi$
$239$	−9.31462	−0.602513	−0.301256	+	0.953543i	$0.597406\pi$
$240$	0	0
$241$	−8.96831	−0.577700	−0.288850	−	0.957374i	$-0.593273\pi$
$241$	−8.96831	−0.577700	−0.288850	+	0.957374i	$0.593273\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	24.8469	1.58741
$246$	0	0
$247$	8.30632	0.528518
$248$	0	0
$249$	13.8633	0.878550
$250$	0	0
$251$	20.5794	1.29896	0.649480	−	0.760379i	$-0.274987\pi$
$251$	20.5794	1.29896	0.649480	+	0.760379i	$0.274987\pi$
$252$	0	0
$253$	52.2449	3.28461
$254$	0	0
$255$	−13.3255	−0.834476
$256$	0	0
$257$	−28.3369	−1.76761	−0.883803	−	0.467859i	$-0.845025\pi$
$257$	−28.3369	−1.76761	−0.883803	+	0.467859i	$0.845025\pi$
$258$	0	0
$259$	2.90304	0.180386
$260$	0	0
$261$	2.90838	0.180024
$262$	0	0
$263$	21.9168	1.35145	0.675723	−	0.737156i	$-0.263832\pi$
$263$	21.9168	1.35145	0.675723	+	0.737156i	$0.263832\pi$
$264$	0	0
$265$	26.3954	1.62146
$266$	0	0
$267$	5.45522	0.333854
$268$	0	0
$269$	8.34233	0.508641	0.254320	−	0.967120i	$-0.418148\pi$
$269$	8.34233	0.508641	0.254320	+	0.967120i	$0.418148\pi$
$270$	0	0
$271$	−5.97049	−0.362681	−0.181341	−	0.983420i	$-0.558044\pi$
$271$	−5.97049	−0.362681	−0.181341	+	0.983420i	$0.558044\pi$
$272$	0	0
$273$	5.86085	0.354715
$274$	0	0
$275$	−63.3202	−3.81835
$276$	0	0
$277$	−2.35540	−0.141522	−0.0707610	−	0.997493i	$-0.522543\pi$
$277$	−2.35540	−0.141522	−0.0707610	+	0.997493i	$0.522543\pi$
$278$	0	0
$279$	−10.6281	−0.636287
$280$	0	0
$281$	−9.45133	−0.563819	−0.281909	−	0.959441i	$-0.590968\pi$
$281$	−9.45133	−0.563819	−0.281909	+	0.959441i	$0.590968\pi$
$282$	0	0
$283$	9.76734	0.580608	0.290304	−	0.956934i	$-0.406244\pi$
$283$	9.76734	0.580608	0.290304	+	0.956934i	$0.406244\pi$
$284$	0	0
$285$	−5.19293	−0.307603
$286$	0	0
$287$	1.63063	0.0962532
$288$	0	0
$289$	−6.04453	−0.355561
$290$	0	0
$291$	13.0108	0.762709
$292$	0	0
$293$	1.86304	0.108840	0.0544200	−	0.998518i	$-0.482669\pi$
$293$	1.86304	0.108840	0.0544200	+	0.998518i	$0.482669\pi$
$294$	0	0
$295$	11.9726	0.697069
$296$	0	0
$297$	−5.64943	−0.327813
$298$	0	0
$299$	−59.5530	−3.44404
$300$	0	0
$301$	2.32958	0.134275
$302$	0	0
$303$	13.5075	0.775986
$304$	0	0
$305$	−33.6087	−1.92443
$306$	0	0
$307$	11.9878	0.684178	0.342089	−	0.939668i	$-0.388866\pi$
$307$	11.9878	0.684178	0.342089	+	0.939668i	$0.388866\pi$
$308$	0	0
$309$	−2.77840	−0.158057
$310$	0	0
$311$	−7.72229	−0.437891	−0.218946	−	0.975737i	$-0.570262\pi$
$311$	−7.72229	−0.437891	−0.218946	+	0.975737i	$0.570262\pi$
$312$	0	0
$313$	−27.5266	−1.55590	−0.777949	−	0.628328i	$-0.783740\pi$
$313$	−27.5266	−1.55590	−0.777949	+	0.628328i	$0.783740\pi$
$314$	0	0
$315$	−3.66408	−0.206447
$316$	0	0
$317$	−29.6703	−1.66645	−0.833225	−	0.552934i	$-0.813508\pi$
$317$	−29.6703	−1.66645	−0.833225	+	0.552934i	$0.813508\pi$
$318$	0	0
$319$	−16.4307	−0.919940
$320$	0	0
$321$	3.24267	0.180988
$322$	0	0
$323$	4.26933	0.237552
$324$	0	0
$325$	72.1776	4.00369
$326$	0	0
$327$	9.99492	0.552720
$328$	0	0
$329$	1.74450	0.0961776
$330$	0	0
$331$	−9.06789	−0.498416	−0.249208	−	0.968450i	$-0.580170\pi$
$331$	−9.06789	−0.498416	−0.249208	+	0.968450i	$0.580170\pi$
$332$	0	0
$333$	3.18976	0.174798
$334$	0	0
$335$	−17.0687	−0.932563
$336$	0	0
$337$	24.1926	1.31785	0.658927	−	0.752207i	$-0.271011\pi$
$337$	24.1926	1.31785	0.658927	+	0.752207i	$0.271011\pi$
$338$	0	0
$339$	12.1305	0.658838
$340$	0	0
$341$	60.0426	3.25149
$342$	0	0
$343$	−11.9878	−0.647278
$344$	0	0
$345$	37.2313	2.00446
$346$	0	0
$347$	11.3260	0.608011	0.304006	−	0.952670i	$-0.401676\pi$
$347$	11.3260	0.608011	0.304006	+	0.952670i	$0.401676\pi$
$348$	0	0
$349$	−30.8336	−1.65048	−0.825241	−	0.564781i	$-0.808961\pi$
$349$	−30.8336	−1.65048	−0.825241	+	0.564781i	$0.808961\pi$
$350$	0	0
$351$	6.43968	0.343725
$352$	0	0
$353$	−29.5939	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$353$	−29.5939	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$354$	0	0
$355$	−40.7266	−2.16155
$356$	0	0
$357$	3.01239	0.159433
$358$	0	0
$359$	−11.0055	−0.580847	−0.290423	−	0.956898i	$-0.593796\pi$
$359$	−11.0055	−0.580847	−0.290423	+	0.956898i	$0.593796\pi$
$360$	0	0
$361$	−17.3362	−0.912434
$362$	0	0
$363$	20.9160	1.09781
$364$	0	0
$365$	−5.13884	−0.268979
$366$	0	0
$367$	−22.8033	−1.19032	−0.595161	−	0.803606i	$-0.702912\pi$
$367$	−22.8033	−1.19032	−0.595161	+	0.803606i	$0.702912\pi$
$368$	0	0
$369$	1.79168	0.0932711
$370$	0	0
$371$	−5.96700	−0.309791
$372$	0	0
$373$	−6.84106	−0.354217	−0.177108	−	0.984191i	$-0.556674\pi$
$373$	−6.84106	−0.354217	−0.177108	+	0.984191i	$0.556674\pi$
$374$	0	0
$375$	−24.9941	−1.29069
$376$	0	0
$377$	18.7290	0.964593
$378$	0	0
$379$	16.7773	0.861789	0.430895	−	0.902402i	$-0.358198\pi$
$379$	16.7773	0.861789	0.430895	+	0.902402i	$0.358198\pi$
$380$	0	0
$381$	12.6803	0.649633
$382$	0	0
$383$	−17.9407	−0.916726	−0.458363	−	0.888765i	$-0.651564\pi$
$383$	−17.9407	−0.916726	−0.458363	+	0.888765i	$0.651564\pi$
$384$	0	0
$385$	20.6999	1.05497
$386$	0	0
$387$	2.55965	0.130115
$388$	0	0
$389$	15.8167	0.801939	0.400970	−	0.916091i	$-0.368673\pi$
$389$	15.8167	0.801939	0.400970	+	0.916091i	$0.368673\pi$
$390$	0	0
$391$	−30.6094	−1.54798
$392$	0	0
$393$	7.08979	0.357633
$394$	0	0
$395$	7.21506	0.363029
$396$	0	0
$397$	3.10390	0.155781	0.0778903	−	0.996962i	$-0.475182\pi$
$397$	3.10390	0.155781	0.0778903	+	0.996962i	$0.475182\pi$
$398$	0	0
$399$	1.17393	0.0587698
$400$	0	0
$401$	6.30176	0.314695	0.157348	−	0.987543i	$-0.449706\pi$
$401$	6.30176	0.314695	0.157348	+	0.987543i	$0.449706\pi$
$402$	0	0
$403$	−68.4415	−3.40931
$404$	0	0
$405$	−4.02595	−0.200051
$406$	0	0
$407$	−18.0203	−0.893233
$408$	0	0
$409$	16.0912	0.795658	0.397829	−	0.917460i	$-0.369764\pi$
$409$	16.0912	0.795658	0.397829	+	0.917460i	$0.369764\pi$
$410$	0	0
$411$	1.44562	0.0713071
$412$	0	0
$413$	−2.70654	−0.133180
$414$	0	0
$415$	−55.8129	−2.73975
$416$	0	0
$417$	−19.4215	−0.951076
$418$	0	0
$419$	28.0306	1.36938	0.684691	−	0.728833i	$-0.259937\pi$
$419$	28.0306	1.36938	0.684691	+	0.728833i	$0.259937\pi$
$420$	0	0
$421$	3.41757	0.166562	0.0832811	−	0.996526i	$-0.473460\pi$
$421$	3.41757	0.166562	0.0832811	+	0.996526i	$0.473460\pi$
$422$	0	0
$423$	1.91679	0.0931977
$424$	0	0
$425$	37.0983	1.79953
$426$	0	0
$427$	7.59767	0.367677
$428$	0	0
$429$	−36.3805	−1.75647
$430$	0	0
$431$	5.43379	0.261737	0.130868	−	0.991400i	$-0.458223\pi$
$431$	5.43379	0.261737	0.130868	+	0.991400i	$0.458223\pi$
$432$	0	0
$433$	40.9533	1.96809	0.984044	−	0.177923i	$-0.0569378\pi$
$433$	40.9533	1.96809	0.984044	+	0.177923i	$0.0569378\pi$
$434$	0	0
$435$	−11.7090	−0.561402
$436$	0	0
$437$	−11.9284	−0.570615
$438$	0	0
$439$	−11.0418	−0.526994	−0.263497	−	0.964660i	$-0.584876\pi$
$439$	−11.0418	−0.526994	−0.263497	+	0.964660i	$0.584876\pi$
$440$	0	0
$441$	−6.17169	−0.293890
$442$	0	0
$443$	−2.25699	−0.107233	−0.0536164	−	0.998562i	$-0.517075\pi$
$443$	−2.25699	−0.107233	−0.0536164	+	0.998562i	$0.517075\pi$
$444$	0	0
$445$	−21.9624	−1.04112
$446$	0	0
$447$	10.6988	0.506037
$448$	0	0
$449$	−34.4263	−1.62468	−0.812339	−	0.583185i	$-0.801806\pi$
$449$	−34.4263	−1.62468	−0.812339	+	0.583185i	$0.801806\pi$
$450$	0	0
$451$	−10.1220	−0.476624
$452$	0	0
$453$	1.13414	0.0532865
$454$	0	0
$455$	−23.5955	−1.10617
$456$	0	0
$457$	5.47846	0.256271	0.128136	−	0.991757i	$-0.459101\pi$
$457$	5.47846	0.256271	0.128136	+	0.991757i	$0.459101\pi$
$458$	0	0
$459$	3.30990	0.154493
$460$	0	0
$461$	21.7152	1.01138	0.505689	−	0.862716i	$-0.331238\pi$
$461$	21.7152	1.01138	0.505689	+	0.862716i	$0.331238\pi$
$462$	0	0
$463$	4.49087	0.208709	0.104354	−	0.994540i	$-0.466722\pi$
$463$	4.49087	0.208709	0.104354	+	0.994540i	$0.466722\pi$
$464$	0	0
$465$	42.7882	1.98425
$466$	0	0
$467$	0.0664030	0.00307277	0.00153638	−	0.999999i	$-0.499511\pi$
$467$	0.0664030	0.00307277	0.00153638	+	0.999999i	$0.499511\pi$
$468$	0	0
$469$	3.85859	0.178173
$470$	0	0
$471$	−2.13747	−0.0984895
$472$	0	0
$473$	−14.4606	−0.664898
$474$	0	0
$475$	14.4571	0.663339
$476$	0	0
$477$	−6.55631	−0.300193
$478$	0	0
$479$	28.4840	1.30147	0.650733	−	0.759307i	$-0.274462\pi$
$479$	28.4840	1.30147	0.650733	+	0.759307i	$0.274462\pi$
$480$	0	0
$481$	20.5410	0.936589
$482$	0	0
$483$	−8.41658	−0.382968
$484$	0	0
$485$	−52.3810	−2.37850
$486$	0	0
$487$	−39.2150	−1.77700	−0.888502	−	0.458873i	$-0.848253\pi$
$487$	−39.2150	−1.77700	−0.888502	+	0.458873i	$0.848253\pi$
$488$	0	0
$489$	−8.75643	−0.395980
$490$	0	0
$491$	25.5329	1.15228	0.576141	−	0.817350i	$-0.304558\pi$
$491$	25.5329	1.15228	0.576141	+	0.817350i	$0.304558\pi$
$492$	0	0
$493$	9.62645	0.433554
$494$	0	0
$495$	22.7443	1.02228
$496$	0	0
$497$	9.20675	0.412979
$498$	0	0
$499$	25.9599	1.16212	0.581061	−	0.813860i	$-0.302638\pi$
$499$	25.9599	1.16212	0.581061	+	0.813860i	$0.302638\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−34.3605	−1.53206	−0.766030	−	0.642805i	$-0.777770\pi$
$503$	−34.3605	−1.53206	−0.766030	+	0.642805i	$0.777770\pi$
$504$	0	0
$505$	−54.3805	−2.41990
$506$	0	0
$507$	28.4695	1.26437
$508$	0	0
$509$	18.1390	0.803998	0.401999	−	0.915640i	$-0.368315\pi$
$509$	18.1390	0.803998	0.401999	+	0.915640i	$0.368315\pi$
$510$	0	0
$511$	1.16170	0.0513905
$512$	0	0
$513$	1.28987	0.0569489
$514$	0	0
$515$	11.1857	0.492900
$516$	0	0
$517$	−10.8288	−0.476250
$518$	0	0
$519$	20.7771	0.912016
$520$	0	0
$521$	−18.7860	−0.823031	−0.411515	−	0.911403i	$-0.635000\pi$
$521$	−18.7860	−0.823031	−0.411515	+	0.911403i	$0.635000\pi$
$522$	0	0
$523$	−2.94829	−0.128920	−0.0644600	−	0.997920i	$-0.520532\pi$
$523$	−2.94829	−0.128920	−0.0644600	+	0.997920i	$0.520532\pi$
$524$	0	0
$525$	10.2008	0.445200
$526$	0	0
$527$	−35.1780	−1.53238
$528$	0	0
$529$	62.5223	2.71836
$530$	0	0
$531$	−2.97385	−0.129054
$532$	0	0
$533$	11.5378	0.499759
$534$	0	0
$535$	−13.0548	−0.564409
$536$	0	0
$537$	20.6874	0.892726
$538$	0	0
$539$	34.8665	1.50181
$540$	0	0
$541$	11.9263	0.512751	0.256375	−	0.966577i	$-0.417472\pi$
$541$	11.9263	0.512751	0.256375	+	0.966577i	$0.417472\pi$
$542$	0	0
$543$	−10.6867	−0.458609
$544$	0	0
$545$	−40.2390	−1.72365
$546$	0	0
$547$	−8.94329	−0.382387	−0.191194	−	0.981552i	$-0.561236\pi$
$547$	−8.94329	−0.382387	−0.191194	+	0.981552i	$0.561236\pi$
$548$	0	0
$549$	8.34803	0.356285
$550$	0	0
$551$	3.75141	0.159816
$552$	0	0
$553$	−1.63105	−0.0693594
$554$	0	0
$555$	−12.8418	−0.545104
$556$	0	0
$557$	−38.5579	−1.63375	−0.816876	−	0.576814i	$-0.804296\pi$
$557$	−38.5579	−1.63375	−0.816876	+	0.576814i	$0.804296\pi$
$558$	0	0
$559$	16.4834	0.697172
$560$	0	0
$561$	−18.6991	−0.789475
$562$	0	0
$563$	10.8490	0.457232	0.228616	−	0.973517i	$-0.426580\pi$
$563$	10.8490	0.457232	0.228616	+	0.973517i	$0.426580\pi$
$564$	0	0
$565$	−48.8368	−2.05458
$566$	0	0
$567$	0.910115	0.0382212
$568$	0	0
$569$	−2.09232	−0.0877148	−0.0438574	−	0.999038i	$-0.513965\pi$
$569$	−2.09232	−0.0877148	−0.0438574	+	0.999038i	$0.513965\pi$
$570$	0	0
$571$	−20.2241	−0.846352	−0.423176	−	0.906047i	$-0.639085\pi$
$571$	−20.2241	−0.846352	−0.423176	+	0.906047i	$0.639085\pi$
$572$	0	0
$573$	23.7568	0.992453
$574$	0	0
$575$	−103.652	−4.32259
$576$	0	0
$577$	31.9254	1.32907	0.664536	−	0.747256i	$-0.268629\pi$
$577$	31.9254	1.32907	0.664536	+	0.747256i	$0.268629\pi$
$578$	0	0
$579$	0.301545	0.0125318
$580$	0	0
$581$	12.6172	0.523449
$582$	0	0
$583$	37.0394	1.53402
$584$	0	0
$585$	−25.9258	−1.07190
$586$	0	0
$587$	20.2855	0.837274	0.418637	−	0.908154i	$-0.362508\pi$
$587$	20.2855	0.837274	0.418637	+	0.908154i	$0.362508\pi$
$588$	0	0
$589$	−13.7088	−0.564862
$590$	0	0
$591$	−21.9860	−0.904382
$592$	0	0
$593$	15.1257	0.621139	0.310570	−	0.950551i	$-0.399480\pi$
$593$	15.1257	0.621139	0.310570	+	0.950551i	$0.399480\pi$
$594$	0	0
$595$	−12.1277	−0.497189
$596$	0	0
$597$	−18.3274	−0.750091
$598$	0	0
$599$	29.7919	1.21727	0.608633	−	0.793452i	$-0.291718\pi$
$599$	29.7919	1.21727	0.608633	+	0.793452i	$0.291718\pi$
$600$	0	0
$601$	−26.0228	−1.06149	−0.530746	−	0.847531i	$-0.678088\pi$
$601$	−26.0228	−1.06149	−0.530746	+	0.847531i	$0.678088\pi$
$602$	0	0
$603$	4.23967	0.172653
$604$	0	0
$605$	−84.2068	−3.42349
$606$	0	0
$607$	2.81543	0.114275	0.0571373	−	0.998366i	$-0.481803\pi$
$607$	2.81543	0.114275	0.0571373	+	0.998366i	$0.481803\pi$
$608$	0	0
$609$	2.64696	0.107260
$610$	0	0
$611$	12.3435	0.499366
$612$	0	0
$613$	−20.9337	−0.845504	−0.422752	−	0.906245i	$-0.638936\pi$
$613$	−20.9337	−0.845504	−0.422752	+	0.906245i	$0.638936\pi$
$614$	0	0
$615$	−7.21321	−0.290865
$616$	0	0
$617$	17.3534	0.698623	0.349312	−	0.937007i	$-0.386415\pi$
$617$	17.3534	0.698623	0.349312	+	0.937007i	$0.386415\pi$
$618$	0	0
$619$	20.9527	0.842159	0.421079	−	0.907024i	$-0.361651\pi$
$619$	20.9527	0.842159	0.421079	+	0.907024i	$0.361651\pi$
$620$	0	0
$621$	−9.24783	−0.371102
$622$	0	0
$623$	4.96488	0.198914
$624$	0	0
$625$	44.5838	1.78335
$626$	0	0
$627$	−7.28700	−0.291015
$628$	0	0
$629$	10.5578	0.420967
$630$	0	0
$631$	16.6037	0.660983	0.330492	−	0.943809i	$-0.392785\pi$
$631$	16.6037	0.660983	0.330492	+	0.943809i	$0.392785\pi$
$632$	0	0
$633$	25.9537	1.03157
$634$	0	0
$635$	−51.0504	−2.02587
$636$	0	0
$637$	−39.7437	−1.57470
$638$	0	0
$639$	10.1160	0.400184
$640$	0	0
$641$	23.1161	0.913033	0.456516	−	0.889715i	$-0.349097\pi$
$641$	23.1161	0.913033	0.456516	+	0.889715i	$0.349097\pi$
$642$	0	0
$643$	−7.04889	−0.277981	−0.138991	−	0.990294i	$-0.544386\pi$
$643$	−7.04889	−0.277981	−0.138991	+	0.990294i	$0.544386\pi$
$644$	0	0
$645$	−10.3050	−0.405761
$646$	0	0
$647$	−15.6485	−0.615206	−0.307603	−	0.951515i	$-0.599527\pi$
$647$	−15.6485	−0.615206	−0.307603	+	0.951515i	$0.599527\pi$
$648$	0	0
$649$	16.8005	0.659478
$650$	0	0
$651$	−9.67279	−0.379106
$652$	0	0
$653$	−32.5382	−1.27332	−0.636658	−	0.771146i	$-0.719684\pi$
$653$	−32.5382	−1.27332	−0.636658	+	0.771146i	$0.719684\pi$
$654$	0	0
$655$	−28.5431	−1.11527
$656$	0	0
$657$	1.27643	0.0497983
$658$	0	0
$659$	−7.89404	−0.307508	−0.153754	−	0.988109i	$-0.549136\pi$
$659$	−7.89404	−0.307508	−0.153754	+	0.988109i	$0.549136\pi$
$660$	0	0
$661$	−9.82653	−0.382208	−0.191104	−	0.981570i	$-0.561207\pi$
$661$	−9.82653	−0.382208	−0.191104	+	0.981570i	$0.561207\pi$
$662$	0	0
$663$	21.3147	0.827795
$664$	0	0
$665$	−4.72616	−0.183273
$666$	0	0
$667$	−26.8962	−1.04142
$668$	0	0
$669$	−23.8575	−0.922382
$670$	0	0
$671$	−47.1616	−1.82065
$672$	0	0
$673$	−12.2177	−0.470959	−0.235480	−	0.971879i	$-0.575666\pi$
$673$	−12.2177	−0.470959	−0.235480	+	0.971879i	$0.575666\pi$
$674$	0	0
$675$	11.2083	0.431406
$676$	0	0
$677$	−17.9900	−0.691413	−0.345706	−	0.938343i	$-0.612361\pi$
$677$	−17.9900	−0.691413	−0.345706	+	0.938343i	$0.612361\pi$
$678$	0	0
$679$	11.8414	0.454430
$680$	0	0
$681$	11.8006	0.452200
$682$	0	0
$683$	38.4906	1.47280	0.736401	−	0.676545i	$-0.236524\pi$
$683$	38.4906	1.47280	0.736401	+	0.676545i	$0.236524\pi$
$684$	0	0
$685$	−5.81999	−0.222370
$686$	0	0
$687$	4.93628	0.188331
$688$	0	0
$689$	−42.2206	−1.60848
$690$	0	0
$691$	17.2590	0.656565	0.328283	−	0.944580i	$-0.393530\pi$
$691$	17.2590	0.656565	0.328283	+	0.944580i	$0.393530\pi$
$692$	0	0
$693$	−5.14163	−0.195314
$694$	0	0
$695$	78.1901	2.96592
$696$	0	0
$697$	5.93028	0.224626
$698$	0	0
$699$	20.2292	0.765139
$700$	0	0
$701$	−27.5288	−1.03975	−0.519874	−	0.854243i	$-0.674021\pi$
$701$	−27.5288	−1.03975	−0.519874	+	0.854243i	$0.674021\pi$
$702$	0	0
$703$	4.11436	0.155176
$704$	0	0
$705$	−7.71692	−0.290636
$706$	0	0
$707$	12.2934	0.462340
$708$	0	0
$709$	44.1421	1.65779	0.828895	−	0.559404i	$-0.188970\pi$
$709$	44.1421	1.65779	0.828895	+	0.559404i	$0.188970\pi$
$710$	0	0
$711$	−1.79214	−0.0672104
$712$	0	0
$713$	98.2868	3.68087
$714$	0	0
$715$	146.466	5.47752
$716$	0	0
$717$	−9.31462	−0.347861
$718$	0	0
$719$	20.3261	0.758037	0.379018	−	0.925389i	$-0.376262\pi$
$719$	20.3261	0.758037	0.379018	+	0.925389i	$0.376262\pi$
$720$	0	0
$721$	−2.52866	−0.0941722
$722$	0	0
$723$	−8.96831	−0.333535
$724$	0	0
$725$	32.5979	1.21065
$726$	0	0
$727$	42.3208	1.56959	0.784796	−	0.619754i	$-0.212767\pi$
$727$	42.3208	1.56959	0.784796	+	0.619754i	$0.212767\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.47221	0.313356
$732$	0	0
$733$	−16.9178	−0.624874	−0.312437	−	0.949938i	$-0.601145\pi$
$733$	−16.9178	−0.624874	−0.312437	+	0.949938i	$0.601145\pi$
$734$	0	0
$735$	24.8469	0.916492
$736$	0	0
$737$	−23.9517	−0.882273
$738$	0	0
$739$	7.57315	0.278583	0.139291	−	0.990251i	$-0.455518\pi$
$739$	7.57315	0.278583	0.139291	+	0.990251i	$0.455518\pi$
$740$	0	0
$741$	8.30632	0.305140
$742$	0	0
$743$	−43.7667	−1.60565	−0.802823	−	0.596218i	$-0.796670\pi$
$743$	−43.7667	−1.60565	−0.802823	+	0.596218i	$0.796670\pi$
$744$	0	0
$745$	−43.0729	−1.57807
$746$	0	0
$747$	13.8633	0.507231
$748$	0	0
$749$	2.95120	0.107834
$750$	0	0
$751$	34.8474	1.27160	0.635800	−	0.771854i	$-0.280670\pi$
$751$	34.8474	1.27160	0.635800	+	0.771854i	$0.280670\pi$
$752$	0	0
$753$	20.5794	0.749954
$754$	0	0
$755$	−4.56598	−0.166173
$756$	0	0
$757$	8.95171	0.325356	0.162678	−	0.986679i	$-0.447987\pi$
$757$	8.95171	0.325356	0.162678	+	0.986679i	$0.447987\pi$
$758$	0	0
$759$	52.2449	1.89637
$760$	0	0
$761$	1.90648	0.0691098	0.0345549	−	0.999403i	$-0.488999\pi$
$761$	1.90648	0.0691098	0.0345549	+	0.999403i	$0.488999\pi$
$762$	0	0
$763$	9.09652	0.329316
$764$	0	0
$765$	−13.3255	−0.481785
$766$	0	0
$767$	−19.1506	−0.691489
$768$	0	0
$769$	43.7489	1.57763	0.788813	−	0.614633i	$-0.210696\pi$
$769$	43.7489	1.57763	0.788813	+	0.614633i	$0.210696\pi$
$770$	0	0
$771$	−28.3369	−1.02053
$772$	0	0
$773$	48.8809	1.75812	0.879062	−	0.476708i	$-0.158170\pi$
$773$	48.8809	1.75812	0.879062	+	0.476708i	$0.158170\pi$
$774$	0	0
$775$	−119.122	−4.27900
$776$	0	0
$777$	2.90304	0.104146
$778$	0	0
$779$	2.31102	0.0828010
$780$	0	0
$781$	−57.1498	−2.04498
$782$	0	0
$783$	2.90838	0.103937
$784$	0	0
$785$	8.60535	0.307138
$786$	0	0
$787$	47.8888	1.70705	0.853525	−	0.521052i	$-0.174460\pi$
$787$	47.8888	1.70705	0.853525	+	0.521052i	$0.174460\pi$
$788$	0	0
$789$	21.9168	0.780258
$790$	0	0
$791$	11.0401	0.392542
$792$	0	0
$793$	53.7586	1.90903
$794$	0	0
$795$	26.3954	0.936148
$796$	0	0
$797$	46.4221	1.64436	0.822178	−	0.569231i	$-0.192759\pi$
$797$	46.4221	1.64436	0.822178	+	0.569231i	$0.192759\pi$
$798$	0	0
$799$	6.34441	0.224449
$800$	0	0
$801$	5.45522	0.192751
$802$	0	0
$803$	−7.21109	−0.254474
$804$	0	0
$805$	33.8847	1.19428
$806$	0	0
$807$	8.34233	0.293664
$808$	0	0
$809$	−20.3954	−0.717065	−0.358533	−	0.933517i	$-0.616723\pi$
$809$	−20.3954	−0.717065	−0.358533	+	0.933517i	$0.616723\pi$
$810$	0	0
$811$	−27.1993	−0.955097	−0.477549	−	0.878605i	$-0.658475\pi$
$811$	−27.1993	−0.955097	−0.477549	+	0.878605i	$0.658475\pi$
$812$	0	0
$813$	−5.97049	−0.209394
$814$	0	0
$815$	35.2529	1.23486
$816$	0	0
$817$	3.30161	0.115509
$818$	0	0
$819$	5.86085	0.204795
$820$	0	0
$821$	53.9559	1.88308	0.941538	−	0.336908i	$-0.109381\pi$
$821$	53.9559	1.88308	0.941538	+	0.336908i	$0.109381\pi$
$822$	0	0
$823$	−20.3519	−0.709421	−0.354711	−	0.934976i	$-0.615421\pi$
$823$	−20.3519	−0.709421	−0.354711	+	0.934976i	$0.615421\pi$
$824$	0	0
$825$	−63.3202	−2.20453
$826$	0	0
$827$	−38.5345	−1.33998	−0.669988	−	0.742372i	$-0.733701\pi$
$827$	−38.5345	−1.33998	−0.669988	+	0.742372i	$0.733701\pi$
$828$	0	0
$829$	26.7280	0.928302	0.464151	−	0.885756i	$-0.346360\pi$
$829$	26.7280	0.928302	0.464151	+	0.885756i	$0.346360\pi$
$830$	0	0
$831$	−2.35540	−0.0817077
$832$	0	0
$833$	−20.4277	−0.707778
$834$	0	0
$835$	4.02595	0.139324
$836$	0	0
$837$	−10.6281	−0.367361
$838$	0	0
$839$	−14.9538	−0.516261	−0.258131	−	0.966110i	$-0.583107\pi$
$839$	−14.9538	−0.516261	−0.258131	+	0.966110i	$0.583107\pi$
$840$	0	0
$841$	−20.5413	−0.708322
$842$	0	0
$843$	−9.45133	−0.325521
$844$	0	0
$845$	−114.617	−3.94293
$846$	0	0
$847$	19.0360	0.654084
$848$	0	0
$849$	9.76734	0.335214
$850$	0	0
$851$	−29.4983	−1.01119
$852$	0	0
$853$	3.85973	0.132155	0.0660773	−	0.997815i	$-0.478952\pi$
$853$	3.85973	0.132155	0.0660773	+	0.997815i	$0.478952\pi$
$854$	0	0
$855$	−5.19293	−0.177594
$856$	0	0
$857$	20.1803	0.689347	0.344674	−	0.938723i	$-0.387990\pi$
$857$	20.1803	0.689347	0.344674	+	0.938723i	$0.387990\pi$
$858$	0	0
$859$	−5.19182	−0.177143	−0.0885713	−	0.996070i	$-0.528230\pi$
$859$	−5.19182	−0.177143	−0.0885713	+	0.996070i	$0.528230\pi$
$860$	0	0
$861$	1.63063	0.0555718
$862$	0	0
$863$	−24.6739	−0.839908	−0.419954	−	0.907545i	$-0.637954\pi$
$863$	−24.6739	−0.839908	−0.419954	+	0.907545i	$0.637954\pi$
$864$	0	0
$865$	−83.6477	−2.84411
$866$	0	0
$867$	−6.04453	−0.205283
$868$	0	0
$869$	10.1246	0.343452
$870$	0	0
$871$	27.3021	0.925098
$872$	0	0
$873$	13.0108	0.440350
$874$	0	0
$875$	−22.7475	−0.769007
$876$	0	0
$877$	−21.5120	−0.726409	−0.363205	−	0.931709i	$-0.618317\pi$
$877$	−21.5120	−0.726409	−0.363205	+	0.931709i	$0.618317\pi$
$878$	0	0
$879$	1.86304	0.0628388
$880$	0	0
$881$	7.28798	0.245538	0.122769	−	0.992435i	$-0.460823\pi$
$881$	7.28798	0.245538	0.122769	+	0.992435i	$0.460823\pi$
$882$	0	0
$883$	46.9312	1.57936	0.789681	−	0.613517i	$-0.210246\pi$
$883$	46.9312	1.57936	0.789681	+	0.613517i	$0.210246\pi$
$884$	0	0
$885$	11.9726	0.402453
$886$	0	0
$887$	−14.7405	−0.494939	−0.247469	−	0.968896i	$-0.579599\pi$
$887$	−14.7405	−0.494939	−0.247469	+	0.968896i	$0.579599\pi$
$888$	0	0
$889$	11.5406	0.387058
$890$	0	0
$891$	−5.64943	−0.189263
$892$	0	0
$893$	2.47241	0.0827359
$894$	0	0
$895$	−83.2863	−2.78395
$896$	0	0
$897$	−59.5530	−1.98842
$898$	0	0
$899$	−30.9105	−1.03092
$900$	0	0
$901$	−21.7008	−0.722958
$902$	0	0
$903$	2.32958	0.0775236
$904$	0	0
$905$	43.0240	1.43017
$906$	0	0
$907$	49.1909	1.63336	0.816678	−	0.577093i	$-0.195813\pi$
$907$	49.1909	1.63336	0.816678	+	0.577093i	$0.195813\pi$
$908$	0	0
$909$	13.5075	0.448016
$910$	0	0
$911$	−24.1824	−0.801198	−0.400599	−	0.916254i	$-0.631198\pi$
$911$	−24.1824	−0.801198	−0.400599	+	0.916254i	$0.631198\pi$
$912$	0	0
$913$	−78.3196	−2.59200
$914$	0	0
$915$	−33.6087	−1.11107
$916$	0	0
$917$	6.45253	0.213081
$918$	0	0
$919$	−47.1032	−1.55379	−0.776895	−	0.629630i	$-0.783207\pi$
$919$	−47.1032	−1.55379	−0.776895	+	0.629630i	$0.783207\pi$
$920$	0	0
$921$	11.9878	0.395010
$922$	0	0
$923$	65.1440	2.14424
$924$	0	0
$925$	35.7516	1.17551
$926$	0	0
$927$	−2.77840	−0.0912545
$928$	0	0
$929$	−23.7341	−0.778690	−0.389345	−	0.921092i	$-0.627299\pi$
$929$	−23.7341	−0.778690	−0.389345	+	0.921092i	$0.627299\pi$
$930$	0	0
$931$	−7.96065	−0.260900
$932$	0	0
$933$	−7.72229	−0.252817
$934$	0	0
$935$	75.2814	2.46197
$936$	0	0
$937$	49.3550	1.61236	0.806179	−	0.591672i	$-0.201532\pi$
$937$	49.3550	1.61236	0.806179	+	0.591672i	$0.201532\pi$
$938$	0	0
$939$	−27.5266	−0.898298
$940$	0	0
$941$	−11.4267	−0.372499	−0.186249	−	0.982502i	$-0.559633\pi$
$941$	−11.4267	−0.372499	−0.186249	+	0.982502i	$0.559633\pi$
$942$	0	0
$943$	−16.5691	−0.539565
$944$	0	0
$945$	−3.66408	−0.119192
$946$	0	0
$947$	−34.8099	−1.13117	−0.565585	−	0.824690i	$-0.691350\pi$
$947$	−34.8099	−1.13117	−0.565585	+	0.824690i	$0.691350\pi$
$948$	0	0
$949$	8.21980	0.266826
$950$	0	0
$951$	−29.6703	−0.962126
$952$	0	0
$953$	−55.3657	−1.79347	−0.896736	−	0.442566i	$-0.854068\pi$
$953$	−55.3657	−1.79347	−0.896736	+	0.442566i	$0.854068\pi$
$954$	0	0
$955$	−95.6435	−3.09495
$956$	0	0
$957$	−16.4307	−0.531128
$958$	0	0
$959$	1.31568	0.0424855
$960$	0	0
$961$	81.9564	2.64375
$962$	0	0
$963$	3.24267	0.104493
$964$	0	0
$965$	−1.21400	−0.0390801
$966$	0	0
$967$	41.9967	1.35052	0.675261	−	0.737579i	$-0.264031\pi$
$967$	41.9967	1.35052	0.675261	+	0.737579i	$0.264031\pi$
$968$	0	0
$969$	4.26933	0.137151
$970$	0	0
$971$	−1.16677	−0.0374435	−0.0187218	−	0.999825i	$-0.505960\pi$
$971$	−1.16677	−0.0374435	−0.0187218	+	0.999825i	$0.505960\pi$
$972$	0	0
$973$	−17.6758	−0.566661
$974$	0	0
$975$	72.1776	2.31153
$976$	0	0
$977$	−52.0060	−1.66382	−0.831909	−	0.554912i	$-0.812752\pi$
$977$	−52.0060	−1.66382	−0.831909	+	0.554912i	$0.812752\pi$
$978$	0	0
$979$	−30.8189	−0.984975
$980$	0	0
$981$	9.99492	0.319113
$982$	0	0
$983$	16.8873	0.538622	0.269311	−	0.963053i	$-0.413204\pi$
$983$	16.8873	0.538622	0.269311	+	0.963053i	$0.413204\pi$
$984$	0	0
$985$	88.5144	2.82030
$986$	0	0
$987$	1.74450	0.0555281
$988$	0	0
$989$	−23.6712	−0.752702
$990$	0	0
$991$	−18.8021	−0.597269	−0.298634	−	0.954368i	$-0.596531\pi$
$991$	−18.8021	−0.597269	−0.298634	+	0.954368i	$0.596531\pi$
$992$	0	0
$993$	−9.06789	−0.287761
$994$	0	0
$995$	73.7853	2.33915
$996$	0	0
$997$	1.11330	0.0352587	0.0176294	−	0.999845i	$-0.494388\pi$
$997$	1.11330	0.0352587	0.0176294	+	0.999845i	$0.494388\pi$
$998$	0	0
$999$	3.18976	0.100919

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.02595 q^{5} +0.910115 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.02595 q^{5} +0.910115 q^{7} +1.00000 q^{9} -5.64943 q^{11} +6.43968 q^{13} -4.02595 q^{15} +3.30990 q^{17} +1.28987 q^{19} +0.910115 q^{21} -9.24783 q^{23} +11.2083 q^{25} +1.00000 q^{27} +2.90838 q^{29} -10.6281 q^{31} -5.64943 q^{33} -3.66408 q^{35} +3.18976 q^{37} +6.43968 q^{39} +1.79168 q^{41} +2.55965 q^{43} -4.02595 q^{45} +1.91679 q^{47} -6.17169 q^{49} +3.30990 q^{51} -6.55631 q^{53} +22.7443 q^{55} +1.28987 q^{57} -2.97385 q^{59} +8.34803 q^{61} +0.910115 q^{63} -25.9258 q^{65} +4.23967 q^{67} -9.24783 q^{69} +10.1160 q^{71} +1.27643 q^{73} +11.2083 q^{75} -5.14163 q^{77} -1.79214 q^{79} +1.00000 q^{81} +13.8633 q^{83} -13.3255 q^{85} +2.90838 q^{87} +5.45522 q^{89} +5.86085 q^{91} -10.6281 q^{93} -5.19293 q^{95} +13.0108 q^{97} -5.64943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more