LMFDB - Embedded newform 6040.2.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 $ x^9 - x^8 - 11x^7 + 9x^6 + 32x^5 - 17x^4 - 27x^3 + 10x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.68790$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.325171 q^{3} +1.00000 q^{5} +2.31586 q^{7} -2.89426 q^{9} +O(q^{10})$	$q+0.325171 q^{3} +1.00000 q^{5} +2.31586 q^{7} -2.89426 q^{9} +0.0563199 q^{11} -6.43408 q^{13} +0.325171 q^{15} +3.42872 q^{17} -2.17512 q^{19} +0.753050 q^{21} -3.43828 q^{23} +1.00000 q^{25} -1.91664 q^{27} +9.17899 q^{29} -4.04276 q^{31} +0.0183136 q^{33} +2.31586 q^{35} -2.68712 q^{37} -2.09218 q^{39} +0.966020 q^{41} +3.65204 q^{43} -2.89426 q^{45} +9.41359 q^{47} -1.63680 q^{49} +1.11492 q^{51} -3.66245 q^{53} +0.0563199 q^{55} -0.707288 q^{57} -8.00799 q^{59} +0.207095 q^{61} -6.70271 q^{63} -6.43408 q^{65} -3.39983 q^{67} -1.11803 q^{69} -5.80306 q^{71} -6.19683 q^{73} +0.325171 q^{75} +0.130429 q^{77} -13.8221 q^{79} +8.05955 q^{81} -6.06544 q^{83} +3.42872 q^{85} +2.98474 q^{87} -5.64338 q^{89} -14.9004 q^{91} -1.31459 q^{93} -2.17512 q^{95} -7.46595 q^{97} -0.163005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * 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
$3$	0.325171	0.187738	0.0938688	−	0.995585i	$-0.470077\pi$
$3$	0.325171	0.187738	0.0938688	+	0.995585i	$0.470077\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.31586	0.875313	0.437656	−	0.899142i	$-0.355809\pi$
$7$	2.31586	0.875313	0.437656	+	0.899142i	$0.355809\pi$
$8$	0	0
$9$	−2.89426	−0.964755
$10$	0	0
$11$	0.0563199	0.0169811	0.00849054	−	0.999964i	$-0.497297\pi$
$11$	0.0563199	0.0169811	0.00849054	+	0.999964i	$0.497297\pi$
$12$	0	0
$13$	−6.43408	−1.78449	−0.892246	−	0.451550i	$-0.850871\pi$
$13$	−6.43408	−1.78449	−0.892246	+	0.451550i	$0.850871\pi$
$14$	0	0
$15$	0.325171	0.0839588
$16$	0	0
$17$	3.42872	0.831586	0.415793	−	0.909459i	$-0.363504\pi$
$17$	3.42872	0.831586	0.415793	+	0.909459i	$0.363504\pi$
$18$	0	0
$19$	−2.17512	−0.499008	−0.249504	−	0.968374i	$-0.580268\pi$
$19$	−2.17512	−0.499008	−0.249504	+	0.968374i	$0.580268\pi$
$20$	0	0
$21$	0.753050	0.164329
$22$	0	0
$23$	−3.43828	−0.716931	−0.358466	−	0.933543i	$-0.616700\pi$
$23$	−3.43828	−0.716931	−0.358466	+	0.933543i	$0.616700\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.91664	−0.368858
$28$	0	0
$29$	9.17899	1.70450	0.852248	−	0.523138i	$-0.175239\pi$
$29$	9.17899	1.70450	0.852248	+	0.523138i	$0.175239\pi$
$30$	0	0
$31$	−4.04276	−0.726101	−0.363050	−	0.931769i	$-0.618265\pi$
$31$	−4.04276	−0.726101	−0.363050	+	0.931769i	$0.618265\pi$
$32$	0	0
$33$	0.0183136	0.00318799
$34$	0	0
$35$	2.31586	0.391452
$36$	0	0
$37$	−2.68712	−0.441760	−0.220880	−	0.975301i	$-0.570893\pi$
$37$	−2.68712	−0.441760	−0.220880	+	0.975301i	$0.570893\pi$
$38$	0	0
$39$	−2.09218	−0.335016
$40$	0	0
$41$	0.966020	0.150867	0.0754335	−	0.997151i	$-0.475966\pi$
$41$	0.966020	0.150867	0.0754335	+	0.997151i	$0.475966\pi$
$42$	0	0
$43$	3.65204	0.556931	0.278466	−	0.960446i	$-0.410174\pi$
$43$	3.65204	0.556931	0.278466	+	0.960446i	$0.410174\pi$
$44$	0	0
$45$	−2.89426	−0.431451
$46$	0	0
$47$	9.41359	1.37311	0.686557	−	0.727076i	$-0.259121\pi$
$47$	9.41359	1.37311	0.686557	+	0.727076i	$0.259121\pi$
$48$	0	0
$49$	−1.63680	−0.233828
$50$	0	0
$51$	1.11492	0.156120
$52$	0	0
$53$	−3.66245	−0.503076	−0.251538	−	0.967847i	$-0.580936\pi$
$53$	−3.66245	−0.503076	−0.251538	+	0.967847i	$0.580936\pi$
$54$	0	0
$55$	0.0563199	0.00759417
$56$	0	0
$57$	−0.707288	−0.0936825
$58$	0	0
$59$	−8.00799	−1.04255	−0.521276	−	0.853388i	$-0.674544\pi$
$59$	−8.00799	−1.04255	−0.521276	+	0.853388i	$0.674544\pi$
$60$	0	0
$61$	0.207095	0.0265158	0.0132579	−	0.999912i	$-0.495780\pi$
$61$	0.207095	0.0265158	0.0132579	+	0.999912i	$0.495780\pi$
$62$	0	0
$63$	−6.70271	−0.844462
$64$	0	0
$65$	−6.43408	−0.798049
$66$	0	0
$67$	−3.39983	−0.415356	−0.207678	−	0.978197i	$-0.566591\pi$
$67$	−3.39983	−0.415356	−0.207678	+	0.978197i	$0.566591\pi$
$68$	0	0
$69$	−1.11803	−0.134595
$70$	0	0
$71$	−5.80306	−0.688696	−0.344348	−	0.938842i	$-0.611900\pi$
$71$	−5.80306	−0.688696	−0.344348	+	0.938842i	$0.611900\pi$
$72$	0	0
$73$	−6.19683	−0.725284	−0.362642	−	0.931928i	$-0.618125\pi$
$73$	−6.19683	−0.725284	−0.362642	+	0.931928i	$0.618125\pi$
$74$	0	0
$75$	0.325171	0.0375475
$76$	0	0
$77$	0.130429	0.0148637
$78$	0	0
$79$	−13.8221	−1.55511	−0.777554	−	0.628816i	$-0.783540\pi$
$79$	−13.8221	−1.55511	−0.777554	+	0.628816i	$0.783540\pi$
$80$	0	0
$81$	8.05955	0.895506
$82$	0	0
$83$	−6.06544	−0.665769	−0.332884	−	0.942968i	$-0.608022\pi$
$83$	−6.06544	−0.665769	−0.332884	+	0.942968i	$0.608022\pi$
$84$	0	0
$85$	3.42872	0.371897
$86$	0	0
$87$	2.98474	0.319998
$88$	0	0
$89$	−5.64338	−0.598197	−0.299099	−	0.954222i	$-0.596686\pi$
$89$	−5.64338	−0.598197	−0.299099	+	0.954222i	$0.596686\pi$
$90$	0	0
$91$	−14.9004	−1.56199
$92$	0	0
$93$	−1.31459	−0.136316
$94$	0	0
$95$	−2.17512	−0.223163
$96$	0	0
$97$	−7.46595	−0.758052	−0.379026	−	0.925386i	$-0.623741\pi$
$97$	−7.46595	−0.758052	−0.379026	+	0.925386i	$0.623741\pi$
$98$	0	0
$99$	−0.163005	−0.0163826
$100$	0	0
$101$	−2.32488	−0.231334	−0.115667	−	0.993288i	$-0.536901\pi$
$101$	−2.32488	−0.231334	−0.115667	+	0.993288i	$0.536901\pi$
$102$	0	0
$103$	−5.82507	−0.573961	−0.286981	−	0.957936i	$-0.592652\pi$
$103$	−5.82507	−0.573961	−0.286981	+	0.957936i	$0.592652\pi$
$104$	0	0
$105$	0.753050	0.0734902
$106$	0	0
$107$	5.39889	0.521930	0.260965	−	0.965348i	$-0.415959\pi$
$107$	5.39889	0.521930	0.260965	+	0.965348i	$0.415959\pi$
$108$	0	0
$109$	−2.01104	−0.192623	−0.0963115	−	0.995351i	$-0.530705\pi$
$109$	−2.01104	−0.192623	−0.0963115	+	0.995351i	$0.530705\pi$
$110$	0	0
$111$	−0.873775	−0.0829350
$112$	0	0
$113$	0.0960389	0.00903458	0.00451729	−	0.999990i	$-0.498562\pi$
$113$	0.0960389	0.00903458	0.00451729	+	0.999990i	$0.498562\pi$
$114$	0	0
$115$	−3.43828	−0.320621
$116$	0	0
$117$	18.6219	1.72160
$118$	0	0
$119$	7.94043	0.727898
$120$	0	0
$121$	−10.9968	−0.999712
$122$	0	0
$123$	0.314122	0.0283234
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.76321	−0.688874	−0.344437	−	0.938809i	$-0.611930\pi$
$127$	−7.76321	−0.688874	−0.344437	+	0.938809i	$0.611930\pi$
$128$	0	0
$129$	1.18754	0.104557
$130$	0	0
$131$	7.22872	0.631576	0.315788	−	0.948830i	$-0.397731\pi$
$131$	7.22872	0.631576	0.315788	+	0.948830i	$0.397731\pi$
$132$	0	0
$133$	−5.03728	−0.436788
$134$	0	0
$135$	−1.91664	−0.164958
$136$	0	0
$137$	−13.5309	−1.15603	−0.578013	−	0.816028i	$-0.696172\pi$
$137$	−13.5309	−1.15603	−0.578013	+	0.816028i	$0.696172\pi$
$138$	0	0
$139$	−12.6203	−1.07044	−0.535221	−	0.844712i	$-0.679771\pi$
$139$	−12.6203	−1.07044	−0.535221	+	0.844712i	$0.679771\pi$
$140$	0	0
$141$	3.06103	0.257785
$142$	0	0
$143$	−0.362366	−0.0303026
$144$	0	0
$145$	9.17899	0.762274
$146$	0	0
$147$	−0.532238	−0.0438983
$148$	0	0
$149$	−1.62093	−0.132792	−0.0663958	−	0.997793i	$-0.521150\pi$
$149$	−1.62093	−0.132792	−0.0663958	+	0.997793i	$0.521150\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−9.92361	−0.802277
$154$	0	0
$155$	−4.04276	−0.324722
$156$	0	0
$157$	18.2972	1.46028	0.730139	−	0.683298i	$-0.239455\pi$
$157$	18.2972	1.46028	0.730139	+	0.683298i	$0.239455\pi$
$158$	0	0
$159$	−1.19092	−0.0944464
$160$	0	0
$161$	−7.96257	−0.627539
$162$	0	0
$163$	−9.80387	−0.767898	−0.383949	−	0.923354i	$-0.625436\pi$
$163$	−9.80387	−0.767898	−0.383949	+	0.923354i	$0.625436\pi$
$164$	0	0
$165$	0.0183136	0.00142571
$166$	0	0
$167$	−2.51591	−0.194687	−0.0973434	−	0.995251i	$-0.531035\pi$
$167$	−2.51591	−0.194687	−0.0973434	+	0.995251i	$0.531035\pi$
$168$	0	0
$169$	28.3973	2.18441
$170$	0	0
$171$	6.29538	0.481420
$172$	0	0
$173$	−20.9962	−1.59631	−0.798156	−	0.602450i	$-0.794191\pi$
$173$	−20.9962	−1.59631	−0.798156	+	0.602450i	$0.794191\pi$
$174$	0	0
$175$	2.31586	0.175063
$176$	0	0
$177$	−2.60397	−0.195726
$178$	0	0
$179$	7.11164	0.531549	0.265774	−	0.964035i	$-0.414372\pi$
$179$	7.11164	0.531549	0.265774	+	0.964035i	$0.414372\pi$
$180$	0	0
$181$	−9.61795	−0.714897	−0.357448	−	0.933933i	$-0.616353\pi$
$181$	−9.61795	−0.714897	−0.357448	+	0.933933i	$0.616353\pi$
$182$	0	0
$183$	0.0673413	0.00497801
$184$	0	0
$185$	−2.68712	−0.197561
$186$	0	0
$187$	0.193105	0.0141212
$188$	0	0
$189$	−4.43868	−0.322866
$190$	0	0
$191$	−6.46910	−0.468087	−0.234044	−	0.972226i	$-0.575196\pi$
$191$	−6.46910	−0.468087	−0.234044	+	0.972226i	$0.575196\pi$
$192$	0	0
$193$	−22.8585	−1.64539	−0.822696	−	0.568482i	$-0.807531\pi$
$193$	−22.8585	−1.64539	−0.822696	+	0.568482i	$0.807531\pi$
$194$	0	0
$195$	−2.09218	−0.149824
$196$	0	0
$197$	−13.2245	−0.942206	−0.471103	−	0.882078i	$-0.656144\pi$
$197$	−13.2245	−0.942206	−0.471103	+	0.882078i	$0.656144\pi$
$198$	0	0
$199$	15.9888	1.13342	0.566709	−	0.823918i	$-0.308216\pi$
$199$	15.9888	1.13342	0.566709	+	0.823918i	$0.308216\pi$
$200$	0	0
$201$	−1.10553	−0.0779779
$202$	0	0
$203$	21.2573	1.49197
$204$	0	0
$205$	0.966020	0.0674698
$206$	0	0
$207$	9.95129	0.691663
$208$	0	0
$209$	−0.122503	−0.00847369
$210$	0	0
$211$	27.6637	1.90445	0.952224	−	0.305401i	$-0.0987905\pi$
$211$	27.6637	1.90445	0.952224	+	0.305401i	$0.0987905\pi$
$212$	0	0
$213$	−1.88699	−0.129294
$214$	0	0
$215$	3.65204	0.249067
$216$	0	0
$217$	−9.36246	−0.635565
$218$	0	0
$219$	−2.01503	−0.136163
$220$	0	0
$221$	−22.0606	−1.48396
$222$	0	0
$223$	−23.4451	−1.57000	−0.784999	−	0.619497i	$-0.787337\pi$
$223$	−23.4451	−1.57000	−0.784999	+	0.619497i	$0.787337\pi$
$224$	0	0
$225$	−2.89426	−0.192951
$226$	0	0
$227$	26.5443	1.76181	0.880905	−	0.473294i	$-0.156935\pi$
$227$	26.5443	1.76181	0.880905	+	0.473294i	$0.156935\pi$
$228$	0	0
$229$	−11.2223	−0.741592	−0.370796	−	0.928714i	$-0.620915\pi$
$229$	−11.2223	−0.741592	−0.370796	+	0.928714i	$0.620915\pi$
$230$	0	0
$231$	0.0424117	0.00279048
$232$	0	0
$233$	6.19435	0.405805	0.202903	−	0.979199i	$-0.434963\pi$
$233$	6.19435	0.405805	0.202903	+	0.979199i	$0.434963\pi$
$234$	0	0
$235$	9.41359	0.614075
$236$	0	0
$237$	−4.49455	−0.291952
$238$	0	0
$239$	17.0596	1.10349	0.551747	−	0.834012i	$-0.313962\pi$
$239$	17.0596	1.10349	0.551747	+	0.834012i	$0.313962\pi$
$240$	0	0
$241$	−10.0660	−0.648408	−0.324204	−	0.945987i	$-0.605096\pi$
$241$	−10.0660	−0.648408	−0.324204	+	0.945987i	$0.605096\pi$
$242$	0	0
$243$	8.37067	0.536978
$244$	0	0
$245$	−1.63680	−0.104571
$246$	0	0
$247$	13.9949	0.890475
$248$	0	0
$249$	−1.97231	−0.124990
$250$	0	0
$251$	2.20064	0.138903	0.0694516	−	0.997585i	$-0.477875\pi$
$251$	2.20064	0.138903	0.0694516	+	0.997585i	$0.477875\pi$
$252$	0	0
$253$	−0.193643	−0.0121743
$254$	0	0
$255$	1.11492	0.0698190
$256$	0	0
$257$	−4.42680	−0.276136	−0.138068	−	0.990423i	$-0.544089\pi$
$257$	−4.42680	−0.276136	−0.138068	+	0.990423i	$0.544089\pi$
$258$	0	0
$259$	−6.22300	−0.386678
$260$	0	0
$261$	−26.5664	−1.64442
$262$	0	0
$263$	16.1049	0.993071	0.496535	−	0.868016i	$-0.334605\pi$
$263$	16.1049	0.993071	0.496535	+	0.868016i	$0.334605\pi$
$264$	0	0
$265$	−3.66245	−0.224983
$266$	0	0
$267$	−1.83506	−0.112304
$268$	0	0
$269$	6.82264	0.415984	0.207992	−	0.978131i	$-0.433307\pi$
$269$	6.82264	0.415984	0.207992	+	0.978131i	$0.433307\pi$
$270$	0	0
$271$	0.306947	0.0186457	0.00932284	−	0.999957i	$-0.497032\pi$
$271$	0.306947	0.0186457	0.00932284	+	0.999957i	$0.497032\pi$
$272$	0	0
$273$	−4.84518	−0.293244
$274$	0	0
$275$	0.0563199	0.00339622
$276$	0	0
$277$	16.0066	0.961741	0.480871	−	0.876792i	$-0.340321\pi$
$277$	16.0066	0.961741	0.480871	+	0.876792i	$0.340321\pi$
$278$	0	0
$279$	11.7008	0.700509
$280$	0	0
$281$	−7.87401	−0.469724	−0.234862	−	0.972029i	$-0.575464\pi$
$281$	−7.87401	−0.469724	−0.234862	+	0.972029i	$0.575464\pi$
$282$	0	0
$283$	−3.03592	−0.180467	−0.0902335	−	0.995921i	$-0.528761\pi$
$283$	−3.03592	−0.180467	−0.0902335	+	0.995921i	$0.528761\pi$
$284$	0	0
$285$	−0.707288	−0.0418961
$286$	0	0
$287$	2.23717	0.132056
$288$	0	0
$289$	−5.24389	−0.308464
$290$	0	0
$291$	−2.42771	−0.142315
$292$	0	0
$293$	0.769903	0.0449782	0.0224891	−	0.999747i	$-0.492841\pi$
$293$	0.769903	0.0449782	0.0224891	+	0.999747i	$0.492841\pi$
$294$	0	0
$295$	−8.00799	−0.466243
$296$	0	0
$297$	−0.107945	−0.00626361
$298$	0	0
$299$	22.1222	1.27936
$300$	0	0
$301$	8.45761	0.487489
$302$	0	0
$303$	−0.755984	−0.0434302
$304$	0	0
$305$	0.207095	0.0118582
$306$	0	0
$307$	8.91000	0.508521	0.254260	−	0.967136i	$-0.418168\pi$
$307$	8.91000	0.508521	0.254260	+	0.967136i	$0.418168\pi$
$308$	0	0
$309$	−1.89414	−0.107754
$310$	0	0
$311$	1.97993	0.112271	0.0561357	−	0.998423i	$-0.482122\pi$
$311$	1.97993	0.112271	0.0561357	+	0.998423i	$0.482122\pi$
$312$	0	0
$313$	1.28314	0.0725271	0.0362636	−	0.999342i	$-0.488454\pi$
$313$	1.28314	0.0725271	0.0362636	+	0.999342i	$0.488454\pi$
$314$	0	0
$315$	−6.70271	−0.377655
$316$	0	0
$317$	−2.47251	−0.138870	−0.0694350	−	0.997586i	$-0.522120\pi$
$317$	−2.47251	−0.138870	−0.0694350	+	0.997586i	$0.522120\pi$
$318$	0	0
$319$	0.516960	0.0289442
$320$	0	0
$321$	1.75556	0.0979859
$322$	0	0
$323$	−7.45789	−0.414968
$324$	0	0
$325$	−6.43408	−0.356898
$326$	0	0
$327$	−0.653933	−0.0361626
$328$	0	0
$329$	21.8006	1.20190
$330$	0	0
$331$	17.9394	0.986037	0.493019	−	0.870019i	$-0.335893\pi$
$331$	17.9394	0.986037	0.493019	+	0.870019i	$0.335893\pi$
$332$	0	0
$333$	7.77724	0.426190
$334$	0	0
$335$	−3.39983	−0.185753
$336$	0	0
$337$	−29.3588	−1.59928	−0.799639	−	0.600481i	$-0.794976\pi$
$337$	−29.3588	−1.59928	−0.799639	+	0.600481i	$0.794976\pi$
$338$	0	0
$339$	0.0312291	0.00169613
$340$	0	0
$341$	−0.227688	−0.0123300
$342$	0	0
$343$	−20.0016	−1.07999
$344$	0	0
$345$	−1.11803	−0.0601927
$346$	0	0
$347$	−6.33413	−0.340034	−0.170017	−	0.985441i	$-0.554382\pi$
$347$	−6.33413	−0.340034	−0.170017	+	0.985441i	$0.554382\pi$
$348$	0	0
$349$	9.10874	0.487580	0.243790	−	0.969828i	$-0.421609\pi$
$349$	9.10874	0.487580	0.243790	+	0.969828i	$0.421609\pi$
$350$	0	0
$351$	12.3318	0.658224
$352$	0	0
$353$	3.92130	0.208710	0.104355	−	0.994540i	$-0.466722\pi$
$353$	3.92130	0.208710	0.104355	+	0.994540i	$0.466722\pi$
$354$	0	0
$355$	−5.80306	−0.307994
$356$	0	0
$357$	2.58200	0.136654
$358$	0	0
$359$	−8.00756	−0.422623	−0.211311	−	0.977419i	$-0.567773\pi$
$359$	−8.00756	−0.422623	−0.211311	+	0.977419i	$0.567773\pi$
$360$	0	0
$361$	−14.2688	−0.750991
$362$	0	0
$363$	−3.57585	−0.187683
$364$	0	0
$365$	−6.19683	−0.324357
$366$	0	0
$367$	8.07193	0.421351	0.210676	−	0.977556i	$-0.432434\pi$
$367$	8.07193	0.421351	0.210676	+	0.977556i	$0.432434\pi$
$368$	0	0
$369$	−2.79592	−0.145550
$370$	0	0
$371$	−8.48172	−0.440349
$372$	0	0
$373$	29.0443	1.50386	0.751928	−	0.659246i	$-0.229124\pi$
$373$	29.0443	1.50386	0.751928	+	0.659246i	$0.229124\pi$
$374$	0	0
$375$	0.325171	0.0167918
$376$	0	0
$377$	−59.0583	−3.04166
$378$	0	0
$379$	−9.84651	−0.505781	−0.252891	−	0.967495i	$-0.581381\pi$
$379$	−9.84651	−0.505781	−0.252891	+	0.967495i	$0.581381\pi$
$380$	0	0
$381$	−2.52437	−0.129327
$382$	0	0
$383$	−1.68356	−0.0860259	−0.0430130	−	0.999075i	$-0.513696\pi$
$383$	−1.68356	−0.0860259	−0.0430130	+	0.999075i	$0.513696\pi$
$384$	0	0
$385$	0.130429	0.00664727
$386$	0	0
$387$	−10.5700	−0.537302
$388$	0	0
$389$	−26.4042	−1.33875	−0.669373	−	0.742926i	$-0.733437\pi$
$389$	−26.4042	−1.33875	−0.669373	+	0.742926i	$0.733437\pi$
$390$	0	0
$391$	−11.7889	−0.596190
$392$	0	0
$393$	2.35057	0.118571
$394$	0	0
$395$	−13.8221	−0.695466
$396$	0	0
$397$	−7.81175	−0.392060	−0.196030	−	0.980598i	$-0.562805\pi$
$397$	−7.81175	−0.392060	−0.196030	+	0.980598i	$0.562805\pi$
$398$	0	0
$399$	−1.63798	−0.0820015
$400$	0	0
$401$	−4.32525	−0.215993	−0.107996	−	0.994151i	$-0.534443\pi$
$401$	−4.32525	−0.215993	−0.107996	+	0.994151i	$0.534443\pi$
$402$	0	0
$403$	26.0114	1.29572
$404$	0	0
$405$	8.05955	0.400482
$406$	0	0
$407$	−0.151338	−0.00750157
$408$	0	0
$409$	32.7052	1.61717	0.808585	−	0.588379i	$-0.200234\pi$
$409$	32.7052	1.61717	0.808585	+	0.588379i	$0.200234\pi$
$410$	0	0
$411$	−4.39987	−0.217029
$412$	0	0
$413$	−18.5454	−0.912559
$414$	0	0
$415$	−6.06544	−0.297741
$416$	0	0
$417$	−4.10376	−0.200962
$418$	0	0
$419$	−10.9558	−0.535225	−0.267613	−	0.963527i	$-0.586235\pi$
$419$	−10.9558	−0.535225	−0.267613	+	0.963527i	$0.586235\pi$
$420$	0	0
$421$	1.67343	0.0815582	0.0407791	−	0.999168i	$-0.487016\pi$
$421$	1.67343	0.0815582	0.0407791	+	0.999168i	$0.487016\pi$
$422$	0	0
$423$	−27.2454	−1.32472
$424$	0	0
$425$	3.42872	0.166317
$426$	0	0
$427$	0.479603	0.0232096
$428$	0	0
$429$	−0.117831	−0.00568893
$430$	0	0
$431$	−32.0330	−1.54297	−0.771487	−	0.636245i	$-0.780487\pi$
$431$	−32.0330	−1.54297	−0.771487	+	0.636245i	$0.780487\pi$
$432$	0	0
$433$	22.2092	1.06731	0.533654	−	0.845703i	$-0.320819\pi$
$433$	22.2092	1.06731	0.533654	+	0.845703i	$0.320819\pi$
$434$	0	0
$435$	2.98474	0.143107
$436$	0	0
$437$	7.47869	0.357754
$438$	0	0
$439$	28.3150	1.35140	0.675700	−	0.737176i	$-0.263841\pi$
$439$	28.3150	1.35140	0.675700	+	0.737176i	$0.263841\pi$
$440$	0	0
$441$	4.73732	0.225587
$442$	0	0
$443$	13.0614	0.620564	0.310282	−	0.950645i	$-0.399577\pi$
$443$	13.0614	0.620564	0.310282	+	0.950645i	$0.399577\pi$
$444$	0	0
$445$	−5.64338	−0.267522
$446$	0	0
$447$	−0.527079	−0.0249300
$448$	0	0
$449$	6.51186	0.307314	0.153657	−	0.988124i	$-0.450895\pi$
$449$	6.51186	0.307314	0.153657	+	0.988124i	$0.450895\pi$
$450$	0	0
$451$	0.0544061	0.00256188
$452$	0	0
$453$	−0.325171	−0.0152779
$454$	0	0
$455$	−14.9004	−0.698542
$456$	0	0
$457$	−14.7695	−0.690890	−0.345445	−	0.938439i	$-0.612272\pi$
$457$	−14.7695	−0.690890	−0.345445	+	0.938439i	$0.612272\pi$
$458$	0	0
$459$	−6.57163	−0.306737
$460$	0	0
$461$	6.06159	0.282317	0.141158	−	0.989987i	$-0.454917\pi$
$461$	6.06159	0.282317	0.141158	+	0.989987i	$0.454917\pi$
$462$	0	0
$463$	31.2815	1.45377	0.726887	−	0.686758i	$-0.240967\pi$
$463$	31.2815	1.45377	0.726887	+	0.686758i	$0.240967\pi$
$464$	0	0
$465$	−1.31459	−0.0609626
$466$	0	0
$467$	11.1705	0.516911	0.258456	−	0.966023i	$-0.416786\pi$
$467$	11.1705	0.516911	0.258456	+	0.966023i	$0.416786\pi$
$468$	0	0
$469$	−7.87354	−0.363566
$470$	0	0
$471$	5.94973	0.274149
$472$	0	0
$473$	0.205682	0.00945729
$474$	0	0
$475$	−2.17512	−0.0998016
$476$	0	0
$477$	10.6001	0.485345
$478$	0	0
$479$	−22.4107	−1.02397	−0.511986	−	0.858994i	$-0.671090\pi$
$479$	−22.4107	−1.02397	−0.511986	+	0.858994i	$0.671090\pi$
$480$	0	0
$481$	17.2892	0.788318
$482$	0	0
$483$	−2.58920	−0.117813
$484$	0	0
$485$	−7.46595	−0.339011
$486$	0	0
$487$	1.70131	0.0770939	0.0385470	−	0.999257i	$-0.487727\pi$
$487$	1.70131	0.0770939	0.0385470	+	0.999257i	$0.487727\pi$
$488$	0	0
$489$	−3.18793	−0.144163
$490$	0	0
$491$	−13.0364	−0.588324	−0.294162	−	0.955756i	$-0.595040\pi$
$491$	−13.0364	−0.588324	−0.294162	+	0.955756i	$0.595040\pi$
$492$	0	0
$493$	31.4722	1.41744
$494$	0	0
$495$	−0.163005	−0.00732651
$496$	0	0
$497$	−13.4391	−0.602825
$498$	0	0
$499$	−20.6803	−0.925780	−0.462890	−	0.886416i	$-0.653187\pi$
$499$	−20.6803	−0.925780	−0.462890	+	0.886416i	$0.653187\pi$
$500$	0	0
$501$	−0.818100	−0.0365500
$502$	0	0
$503$	1.83106	0.0816431	0.0408216	−	0.999166i	$-0.487002\pi$
$503$	1.83106	0.0816431	0.0408216	+	0.999166i	$0.487002\pi$
$504$	0	0
$505$	−2.32488	−0.103456
$506$	0	0
$507$	9.23399	0.410096
$508$	0	0
$509$	−34.4468	−1.52683	−0.763413	−	0.645911i	$-0.776478\pi$
$509$	−34.4468	−1.52683	−0.763413	+	0.645911i	$0.776478\pi$
$510$	0	0
$511$	−14.3510	−0.634851
$512$	0	0
$513$	4.16894	0.184063
$514$	0	0
$515$	−5.82507	−0.256683
$516$	0	0
$517$	0.530172	0.0233169
$518$	0	0
$519$	−6.82736	−0.299688
$520$	0	0
$521$	−11.8029	−0.517097	−0.258548	−	0.965998i	$-0.583244\pi$
$521$	−11.8029	−0.517097	−0.258548	+	0.965998i	$0.583244\pi$
$522$	0	0
$523$	16.3976	0.717015	0.358507	−	0.933527i	$-0.383286\pi$
$523$	16.3976	0.717015	0.358507	+	0.933527i	$0.383286\pi$
$524$	0	0
$525$	0.753050	0.0328658
$526$	0	0
$527$	−13.8615	−0.603816
$528$	0	0
$529$	−11.1782	−0.486010
$530$	0	0
$531$	23.1772	1.00581
$532$	0	0
$533$	−6.21545	−0.269221
$534$	0	0
$535$	5.39889	0.233414
$536$	0	0
$537$	2.31250	0.0997917
$538$	0	0
$539$	−0.0921841	−0.00397065
$540$	0	0
$541$	−21.8874	−0.941012	−0.470506	−	0.882397i	$-0.655929\pi$
$541$	−21.8874	−0.941012	−0.470506	+	0.882397i	$0.655929\pi$
$542$	0	0
$543$	−3.12748	−0.134213
$544$	0	0
$545$	−2.01104	−0.0861437
$546$	0	0
$547$	−24.3436	−1.04086	−0.520429	−	0.853905i	$-0.674228\pi$
$547$	−24.3436	−1.04086	−0.520429	+	0.853905i	$0.674228\pi$
$548$	0	0
$549$	−0.599387	−0.0255812
$550$	0	0
$551$	−19.9655	−0.850557
$552$	0	0
$553$	−32.0101	−1.36121
$554$	0	0
$555$	−0.873775	−0.0370897
$556$	0	0
$557$	−26.3618	−1.11699	−0.558494	−	0.829509i	$-0.688620\pi$
$557$	−26.3618	−1.11699	−0.558494	+	0.829509i	$0.688620\pi$
$558$	0	0
$559$	−23.4975	−0.993839
$560$	0	0
$561$	0.0627921	0.00265109
$562$	0	0
$563$	12.5418	0.528574	0.264287	−	0.964444i	$-0.414863\pi$
$563$	12.5418	0.528574	0.264287	+	0.964444i	$0.414863\pi$
$564$	0	0
$565$	0.0960389	0.00404039
$566$	0	0
$567$	18.6648	0.783848
$568$	0	0
$569$	18.0058	0.754844	0.377422	−	0.926041i	$-0.376811\pi$
$569$	18.0058	0.754844	0.377422	+	0.926041i	$0.376811\pi$
$570$	0	0
$571$	12.7538	0.533731	0.266866	−	0.963734i	$-0.414012\pi$
$571$	12.7538	0.533731	0.266866	+	0.963734i	$0.414012\pi$
$572$	0	0
$573$	−2.10356	−0.0878776
$574$	0	0
$575$	−3.43828	−0.143386
$576$	0	0
$577$	−20.1902	−0.840527	−0.420264	−	0.907402i	$-0.638062\pi$
$577$	−20.1902	−0.840527	−0.420264	+	0.907402i	$0.638062\pi$
$578$	0	0
$579$	−7.43293	−0.308902
$580$	0	0
$581$	−14.0467	−0.582756
$582$	0	0
$583$	−0.206269	−0.00854278
$584$	0	0
$585$	18.6219	0.769921
$586$	0	0
$587$	7.08622	0.292479	0.146240	−	0.989249i	$-0.453283\pi$
$587$	7.08622	0.292479	0.146240	+	0.989249i	$0.453283\pi$
$588$	0	0
$589$	8.79350	0.362330
$590$	0	0
$591$	−4.30022	−0.176887
$592$	0	0
$593$	3.05023	0.125258	0.0626290	−	0.998037i	$-0.480052\pi$
$593$	3.05023	0.125258	0.0626290	+	0.998037i	$0.480052\pi$
$594$	0	0
$595$	7.94043	0.325526
$596$	0	0
$597$	5.19910	0.212785
$598$	0	0
$599$	27.1790	1.11050	0.555252	−	0.831682i	$-0.312622\pi$
$599$	27.1790	1.11050	0.555252	+	0.831682i	$0.312622\pi$
$600$	0	0
$601$	2.03409	0.0829724	0.0414862	−	0.999139i	$-0.486791\pi$
$601$	2.03409	0.0829724	0.0414862	+	0.999139i	$0.486791\pi$
$602$	0	0
$603$	9.84002	0.400716
$604$	0	0
$605$	−10.9968	−0.447085
$606$	0	0
$607$	−8.22738	−0.333939	−0.166969	−	0.985962i	$-0.553398\pi$
$607$	−8.22738	−0.333939	−0.166969	+	0.985962i	$0.553398\pi$
$608$	0	0
$609$	6.91224	0.280098
$610$	0	0
$611$	−60.5678	−2.45031
$612$	0	0
$613$	33.9482	1.37116	0.685578	−	0.727999i	$-0.259549\pi$
$613$	33.9482	1.37116	0.685578	+	0.727999i	$0.259549\pi$
$614$	0	0
$615$	0.314122	0.0126666
$616$	0	0
$617$	−12.3280	−0.496305	−0.248153	−	0.968721i	$-0.579823\pi$
$617$	−12.3280	−0.496305	−0.248153	+	0.968721i	$0.579823\pi$
$618$	0	0
$619$	−20.5233	−0.824901	−0.412450	−	0.910980i	$-0.635327\pi$
$619$	−20.5233	−0.824901	−0.412450	+	0.910980i	$0.635327\pi$
$620$	0	0
$621$	6.58996	0.264446
$622$	0	0
$623$	−13.0693	−0.523609
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.0398343	−0.00159083
$628$	0	0
$629$	−9.21339	−0.367362
$630$	0	0
$631$	14.4763	0.576294	0.288147	−	0.957586i	$-0.406961\pi$
$631$	14.4763	0.576294	0.288147	+	0.957586i	$0.406961\pi$
$632$	0	0
$633$	8.99543	0.357536
$634$	0	0
$635$	−7.76321	−0.308074
$636$	0	0
$637$	10.5313	0.417264
$638$	0	0
$639$	16.7956	0.664423
$640$	0	0
$641$	−28.2070	−1.11411	−0.557055	−	0.830476i	$-0.688069\pi$
$641$	−28.2070	−1.11411	−0.557055	+	0.830476i	$0.688069\pi$
$642$	0	0
$643$	21.1243	0.833059	0.416530	−	0.909122i	$-0.363246\pi$
$643$	21.1243	0.833059	0.416530	+	0.909122i	$0.363246\pi$
$644$	0	0
$645$	1.18754	0.0467593
$646$	0	0
$647$	43.5932	1.71382	0.856912	−	0.515463i	$-0.172380\pi$
$647$	43.5932	1.71382	0.856912	+	0.515463i	$0.172380\pi$
$648$	0	0
$649$	−0.451009	−0.0177037
$650$	0	0
$651$	−3.04440	−0.119319
$652$	0	0
$653$	−42.7553	−1.67314	−0.836572	−	0.547857i	$-0.815444\pi$
$653$	−42.7553	−1.67314	−0.836572	+	0.547857i	$0.815444\pi$
$654$	0	0
$655$	7.22872	0.282449
$656$	0	0
$657$	17.9353	0.699722
$658$	0	0
$659$	40.6170	1.58221	0.791107	−	0.611678i	$-0.209505\pi$
$659$	40.6170	1.58221	0.791107	+	0.611678i	$0.209505\pi$
$660$	0	0
$661$	−47.7230	−1.85621	−0.928105	−	0.372318i	$-0.878563\pi$
$661$	−47.7230	−1.85621	−0.928105	+	0.372318i	$0.878563\pi$
$662$	0	0
$663$	−7.17348	−0.278595
$664$	0	0
$665$	−5.03728	−0.195337
$666$	0	0
$667$	−31.5600	−1.22201
$668$	0	0
$669$	−7.62366	−0.294748
$670$	0	0
$671$	0.0116636	0.000450267 0
$672$	0	0
$673$	25.2841	0.974630	0.487315	−	0.873226i	$-0.337976\pi$
$673$	25.2841	0.974630	0.487315	+	0.873226i	$0.337976\pi$
$674$	0	0
$675$	−1.91664	−0.0737717
$676$	0	0
$677$	32.8260	1.26161	0.630803	−	0.775943i	$-0.282726\pi$
$677$	32.8260	1.26161	0.630803	+	0.775943i	$0.282726\pi$
$678$	0	0
$679$	−17.2901	−0.663532
$680$	0	0
$681$	8.63145	0.330758
$682$	0	0
$683$	−6.92474	−0.264968	−0.132484	−	0.991185i	$-0.542295\pi$
$683$	−6.92474	−0.264968	−0.132484	+	0.991185i	$0.542295\pi$
$684$	0	0
$685$	−13.5309	−0.516990
$686$	0	0
$687$	−3.64917	−0.139225
$688$	0	0
$689$	23.5645	0.897736
$690$	0	0
$691$	−15.7452	−0.598976	−0.299488	−	0.954100i	$-0.596816\pi$
$691$	−15.7452	−0.598976	−0.299488	+	0.954100i	$0.596816\pi$
$692$	0	0
$693$	−0.377496	−0.0143399
$694$	0	0
$695$	−12.6203	−0.478716
$696$	0	0
$697$	3.31221	0.125459
$698$	0	0
$699$	2.01422	0.0761849
$700$	0	0
$701$	20.2907	0.766370	0.383185	−	0.923672i	$-0.374827\pi$
$701$	20.2907	0.766370	0.383185	+	0.923672i	$0.374827\pi$
$702$	0	0
$703$	5.84483	0.220442
$704$	0	0
$705$	3.06103	0.115285
$706$	0	0
$707$	−5.38410	−0.202490
$708$	0	0
$709$	−1.50205	−0.0564106	−0.0282053	−	0.999602i	$-0.508979\pi$
$709$	−1.50205	−0.0564106	−0.0282053	+	0.999602i	$0.508979\pi$
$710$	0	0
$711$	40.0048	1.50030
$712$	0	0
$713$	13.9001	0.520564
$714$	0	0
$715$	−0.362366	−0.0135517
$716$	0	0
$717$	5.54728	0.207167
$718$	0	0
$719$	−16.1931	−0.603900	−0.301950	−	0.953324i	$-0.597638\pi$
$719$	−16.1931	−0.603900	−0.301950	+	0.953324i	$0.597638\pi$
$720$	0	0
$721$	−13.4900	−0.502395
$722$	0	0
$723$	−3.27317	−0.121731
$724$	0	0
$725$	9.17899	0.340899
$726$	0	0
$727$	51.7644	1.91984	0.959918	−	0.280279i	$-0.0904271\pi$
$727$	51.7644	1.91984	0.959918	+	0.280279i	$0.0904271\pi$
$728$	0	0
$729$	−21.4568	−0.794695
$730$	0	0
$731$	12.5218	0.463136
$732$	0	0
$733$	53.4407	1.97388	0.986938	−	0.161102i	$-0.0515048\pi$
$733$	53.4407	1.97388	0.986938	+	0.161102i	$0.0515048\pi$
$734$	0	0
$735$	−0.532238	−0.0196319
$736$	0	0
$737$	−0.191478	−0.00705319
$738$	0	0
$739$	15.9949	0.588382	0.294191	−	0.955747i	$-0.404950\pi$
$739$	15.9949	0.588382	0.294191	+	0.955747i	$0.404950\pi$
$740$	0	0
$741$	4.55074	0.167176
$742$	0	0
$743$	42.3285	1.55288	0.776442	−	0.630189i	$-0.217023\pi$
$743$	42.3285	1.55288	0.776442	+	0.630189i	$0.217023\pi$
$744$	0	0
$745$	−1.62093	−0.0593862
$746$	0	0
$747$	17.5550	0.642303
$748$	0	0
$749$	12.5031	0.456852
$750$	0	0
$751$	−48.2051	−1.75903	−0.879515	−	0.475872i	$-0.842133\pi$
$751$	−48.2051	−1.75903	−0.879515	+	0.475872i	$0.842133\pi$
$752$	0	0
$753$	0.715585	0.0260774
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−19.5650	−0.711101	−0.355550	−	0.934657i	$-0.615707\pi$
$757$	−19.5650	−0.711101	−0.355550	+	0.934657i	$0.615707\pi$
$758$	0	0
$759$	−0.0629673	−0.00228557
$760$	0	0
$761$	10.4368	0.378335	0.189167	−	0.981945i	$-0.439421\pi$
$761$	10.4368	0.378335	0.189167	+	0.981945i	$0.439421\pi$
$762$	0	0
$763$	−4.65730	−0.168605
$764$	0	0
$765$	−9.92361	−0.358789
$766$	0	0
$767$	51.5240	1.86042
$768$	0	0
$769$	−17.3916	−0.627158	−0.313579	−	0.949562i	$-0.601528\pi$
$769$	−17.3916	−0.627158	−0.313579	+	0.949562i	$0.601528\pi$
$770$	0	0
$771$	−1.43947	−0.0518412
$772$	0	0
$773$	−13.8077	−0.496629	−0.248314	−	0.968680i	$-0.579877\pi$
$773$	−13.8077	−0.496629	−0.248314	+	0.968680i	$0.579877\pi$
$774$	0	0
$775$	−4.04276	−0.145220
$776$	0	0
$777$	−2.02354	−0.0725941
$778$	0	0
$779$	−2.10121	−0.0752838
$780$	0	0
$781$	−0.326827	−0.0116948
$782$	0	0
$783$	−17.5929	−0.628717
$784$	0	0
$785$	18.2972	0.653057
$786$	0	0
$787$	36.7967	1.31166	0.655831	−	0.754908i	$-0.272318\pi$
$787$	36.7967	1.31166	0.655831	+	0.754908i	$0.272318\pi$
$788$	0	0
$789$	5.23685	0.186437
$790$	0	0
$791$	0.222413	0.00790808
$792$	0	0
$793$	−1.33246	−0.0473172
$794$	0	0
$795$	−1.19092	−0.0422377
$796$	0	0
$797$	−2.79694	−0.0990726	−0.0495363	−	0.998772i	$-0.515774\pi$
$797$	−2.79694	−0.0990726	−0.0495363	+	0.998772i	$0.515774\pi$
$798$	0	0
$799$	32.2766	1.14186
$800$	0	0
$801$	16.3334	0.577113
$802$	0	0
$803$	−0.349005	−0.0123161
$804$	0	0
$805$	−7.96257	−0.280644
$806$	0	0
$807$	2.21853	0.0780958
$808$	0	0
$809$	−28.5043	−1.00216	−0.501079	−	0.865401i	$-0.667064\pi$
$809$	−28.5043	−1.00216	−0.501079	+	0.865401i	$0.667064\pi$
$810$	0	0
$811$	−38.8251	−1.36333	−0.681666	−	0.731663i	$-0.738744\pi$
$811$	−38.8251	−1.36333	−0.681666	+	0.731663i	$0.738744\pi$
$812$	0	0
$813$	0.0998102	0.00350049
$814$	0	0
$815$	−9.80387	−0.343415
$816$	0	0
$817$	−7.94364	−0.277913
$818$	0	0
$819$	43.1257	1.50694
$820$	0	0
$821$	19.1085	0.666890	0.333445	−	0.942769i	$-0.391789\pi$
$821$	19.1085	0.666890	0.333445	+	0.942769i	$0.391789\pi$
$822$	0	0
$823$	21.7891	0.759521	0.379761	−	0.925085i	$-0.376006\pi$
$823$	21.7891	0.759521	0.379761	+	0.925085i	$0.376006\pi$
$824$	0	0
$825$	0.0183136	0.000637597 0
$826$	0	0
$827$	1.49055	0.0518317	0.0259158	−	0.999664i	$-0.491750\pi$
$827$	1.49055	0.0518317	0.0259158	+	0.999664i	$0.491750\pi$
$828$	0	0
$829$	15.9080	0.552509	0.276254	−	0.961085i	$-0.410907\pi$
$829$	15.9080	0.552509	0.276254	+	0.961085i	$0.410907\pi$
$830$	0	0
$831$	5.20487	0.180555
$832$	0	0
$833$	−5.61211	−0.194448
$834$	0	0
$835$	−2.51591	−0.0870666
$836$	0	0
$837$	7.74853	0.267828
$838$	0	0
$839$	3.58777	0.123864	0.0619318	−	0.998080i	$-0.480274\pi$
$839$	3.58777	0.123864	0.0619318	+	0.998080i	$0.480274\pi$
$840$	0	0
$841$	55.2539	1.90531
$842$	0	0
$843$	−2.56040	−0.0881848
$844$	0	0
$845$	28.3973	0.976898
$846$	0	0
$847$	−25.4671	−0.875060
$848$	0	0
$849$	−0.987194	−0.0338804
$850$	0	0
$851$	9.23908	0.316712
$852$	0	0
$853$	35.0955	1.20165	0.600823	−	0.799382i	$-0.294840\pi$
$853$	35.0955	1.20165	0.600823	+	0.799382i	$0.294840\pi$
$854$	0	0
$855$	6.29538	0.215298
$856$	0	0
$857$	21.8544	0.746534	0.373267	−	0.927724i	$-0.378238\pi$
$857$	21.8544	0.746534	0.373267	+	0.927724i	$0.378238\pi$
$858$	0	0
$859$	15.3375	0.523308	0.261654	−	0.965162i	$-0.415732\pi$
$859$	15.3375	0.523308	0.261654	+	0.965162i	$0.415732\pi$
$860$	0	0
$861$	0.727462	0.0247918
$862$	0	0
$863$	−26.5648	−0.904276	−0.452138	−	0.891948i	$-0.649338\pi$
$863$	−26.5648	−0.904276	−0.452138	+	0.891948i	$0.649338\pi$
$864$	0	0
$865$	−20.9962	−0.713893
$866$	0	0
$867$	−1.70516	−0.0579103
$868$	0	0
$869$	−0.778459	−0.0264074
$870$	0	0
$871$	21.8748	0.741199
$872$	0	0
$873$	21.6084	0.731334
$874$	0	0
$875$	2.31586	0.0782903
$876$	0	0
$877$	9.18843	0.310271	0.155136	−	0.987893i	$-0.450419\pi$
$877$	9.18843	0.310271	0.155136	+	0.987893i	$0.450419\pi$
$878$	0	0
$879$	0.250350	0.00844410
$880$	0	0
$881$	22.6391	0.762731	0.381365	−	0.924424i	$-0.375454\pi$
$881$	22.6391	0.762731	0.381365	+	0.924424i	$0.375454\pi$
$882$	0	0
$883$	−30.3049	−1.01984	−0.509920	−	0.860222i	$-0.670325\pi$
$883$	−30.3049	−1.01984	−0.509920	+	0.860222i	$0.670325\pi$
$884$	0	0
$885$	−2.60397	−0.0875314
$886$	0	0
$887$	17.9430	0.602468	0.301234	−	0.953550i	$-0.402601\pi$
$887$	17.9430	0.602468	0.301234	+	0.953550i	$0.402601\pi$
$888$	0	0
$889$	−17.9785	−0.602980
$890$	0	0
$891$	0.453913	0.0152067
$892$	0	0
$893$	−20.4757	−0.685194
$894$	0	0
$895$	7.11164	0.237716
$896$	0	0
$897$	7.19349	0.240183
$898$	0	0
$899$	−37.1084	−1.23764
$900$	0	0
$901$	−12.5575	−0.418351
$902$	0	0
$903$	2.75017	0.0915200
$904$	0	0
$905$	−9.61795	−0.319712
$906$	0	0
$907$	51.5996	1.71333	0.856667	−	0.515869i	$-0.172531\pi$
$907$	51.5996	1.71333	0.856667	+	0.515869i	$0.172531\pi$
$908$	0	0
$909$	6.72882	0.223181
$910$	0	0
$911$	−28.5238	−0.945036	−0.472518	−	0.881321i	$-0.656655\pi$
$911$	−28.5238	−0.945036	−0.472518	+	0.881321i	$0.656655\pi$
$912$	0	0
$913$	−0.341605	−0.0113055
$914$	0	0
$915$	0.0673413	0.00222623
$916$	0	0
$917$	16.7407	0.552826
$918$	0	0
$919$	36.8287	1.21487	0.607433	−	0.794371i	$-0.292199\pi$
$919$	36.8287	1.21487	0.607433	+	0.794371i	$0.292199\pi$
$920$	0	0
$921$	2.89727	0.0954684
$922$	0	0
$923$	37.3373	1.22897
$924$	0	0
$925$	−2.68712	−0.0883521
$926$	0	0
$927$	16.8593	0.553732
$928$	0	0
$929$	22.0355	0.722961	0.361481	−	0.932380i	$-0.382271\pi$
$929$	22.0355	0.722961	0.361481	+	0.932380i	$0.382271\pi$
$930$	0	0
$931$	3.56023	0.116682
$932$	0	0
$933$	0.643815	0.0210776
$934$	0	0
$935$	0.193105	0.00631521
$936$	0	0
$937$	−22.3699	−0.730792	−0.365396	−	0.930852i	$-0.619066\pi$
$937$	−22.3699	−0.730792	−0.365396	+	0.930852i	$0.619066\pi$
$938$	0	0
$939$	0.417239	0.0136161
$940$	0	0
$941$	−59.0733	−1.92573	−0.962867	−	0.269976i	$-0.912984\pi$
$941$	−59.0733	−1.92573	−0.962867	+	0.269976i	$0.912984\pi$
$942$	0	0
$943$	−3.32145	−0.108161
$944$	0	0
$945$	−4.43868	−0.144390
$946$	0	0
$947$	−20.1454	−0.654639	−0.327320	−	0.944914i	$-0.606145\pi$
$947$	−20.1454	−0.654639	−0.327320	+	0.944914i	$0.606145\pi$
$948$	0	0
$949$	39.8709	1.29426
$950$	0	0
$951$	−0.803988	−0.0260711
$952$	0	0
$953$	−39.6801	−1.28537	−0.642683	−	0.766133i	$-0.722179\pi$
$953$	−39.6801	−1.28537	−0.642683	+	0.766133i	$0.722179\pi$
$954$	0	0
$955$	−6.46910	−0.209335
$956$	0	0
$957$	0.168100	0.00543391
$958$	0	0
$959$	−31.3357	−1.01188
$960$	0	0
$961$	−14.6561	−0.472777
$962$	0	0
$963$	−15.6258	−0.503535
$964$	0	0
$965$	−22.8585	−0.735841
$966$	0	0
$967$	36.5096	1.17407	0.587034	−	0.809562i	$-0.300295\pi$
$967$	36.5096	1.17407	0.587034	+	0.809562i	$0.300295\pi$
$968$	0	0
$969$	−2.42509	−0.0779051
$970$	0	0
$971$	−13.0888	−0.420038	−0.210019	−	0.977697i	$-0.567353\pi$
$971$	−13.0888	−0.420038	−0.210019	+	0.977697i	$0.567353\pi$
$972$	0	0
$973$	−29.2269	−0.936971
$974$	0	0
$975$	−2.09218	−0.0670032
$976$	0	0
$977$	−5.45304	−0.174458	−0.0872291	−	0.996188i	$-0.527801\pi$
$977$	−5.45304	−0.174458	−0.0872291	+	0.996188i	$0.527801\pi$
$978$	0	0
$979$	−0.317834	−0.0101580
$980$	0	0
$981$	5.82049	0.185834
$982$	0	0
$983$	−24.6518	−0.786270	−0.393135	−	0.919481i	$-0.628610\pi$
$983$	−24.6518	−0.786270	−0.393135	+	0.919481i	$0.628610\pi$
$984$	0	0
$985$	−13.2245	−0.421367
$986$	0	0
$987$	7.08891	0.225642
$988$	0	0
$989$	−12.5567	−0.399281
$990$	0	0
$991$	−16.7680	−0.532654	−0.266327	−	0.963883i	$-0.585810\pi$
$991$	−16.7680	−0.532654	−0.266327	+	0.963883i	$0.585810\pi$
$992$	0	0
$993$	5.83337	0.185116
$994$	0	0
$995$	15.9888	0.506880
$996$	0	0
$997$	−4.30387	−0.136305	−0.0681524	−	0.997675i	$-0.521710\pi$
$997$	−4.30387	−0.136305	−0.0681524	+	0.997675i	$0.521710\pi$
$998$	0	0
$999$	5.15026	0.162947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.6	✓	9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.325171 q^{3} +1.00000 q^{5} +2.31586 q^{7} -2.89426 q^{9} +O(q^{10})\)	\(q+0.325171 q^{3} +1.00000 q^{5} +2.31586 q^{7} -2.89426 q^{9} +0.0563199 q^{11} -6.43408 q^{13} +0.325171 q^{15} +3.42872 q^{17} -2.17512 q^{19} +0.753050 q^{21} -3.43828 q^{23} +1.00000 q^{25} -1.91664 q^{27} +9.17899 q^{29} -4.04276 q^{31} +0.0183136 q^{33} +2.31586 q^{35} -2.68712 q^{37} -2.09218 q^{39} +0.966020 q^{41} +3.65204 q^{43} -2.89426 q^{45} +9.41359 q^{47} -1.63680 q^{49} +1.11492 q^{51} -3.66245 q^{53} +0.0563199 q^{55} -0.707288 q^{57} -8.00799 q^{59} +0.207095 q^{61} -6.70271 q^{63} -6.43408 q^{65} -3.39983 q^{67} -1.11803 q^{69} -5.80306 q^{71} -6.19683 q^{73} +0.325171 q^{75} +0.130429 q^{77} -13.8221 q^{79} +8.05955 q^{81} -6.06544 q^{83} +3.42872 q^{85} +2.98474 q^{87} -5.64338 q^{89} -14.9004 q^{91} -1.31459 q^{93} -2.17512 q^{95} -7.46595 q^{97} -0.163005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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