LMFDB - Embedded newform 4008.2.a.e.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.21432 q^{5} -1.37778 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.21432 q^{5} -1.37778 q^{7} +1.00000 q^{9} -3.52543 q^{11} +2.00000 q^{13} -4.21432 q^{15} -6.64296 q^{17} +5.80642 q^{19} +1.37778 q^{21} -8.00000 q^{23} +12.7605 q^{25} -1.00000 q^{27} -9.05086 q^{29} -5.80642 q^{31} +3.52543 q^{33} -5.80642 q^{35} -3.37778 q^{37} -2.00000 q^{39} +10.7763 q^{41} +4.02074 q^{43} +4.21432 q^{45} -6.57628 q^{47} -5.10171 q^{49} +6.64296 q^{51} -9.39853 q^{53} -14.8573 q^{55} -5.80642 q^{57} -3.05086 q^{59} -10.5161 q^{61} -1.37778 q^{63} +8.42864 q^{65} -6.40790 q^{67} +8.00000 q^{69} +8.04149 q^{71} +16.2351 q^{73} -12.7605 q^{75} +4.85728 q^{77} -7.69381 q^{79} +1.00000 q^{81} -14.7971 q^{83} -27.9956 q^{85} +9.05086 q^{87} -6.29529 q^{89} -2.75557 q^{91} +5.80642 q^{93} +24.4701 q^{95} +15.2859 q^{97} -3.52543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 6 q^{15} + 4 q^{19} + 4 q^{21} - 24 q^{23} + 5 q^{25} - 3 q^{27} - 14 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{35} - 10 q^{37} - 6 q^{39} + 12 q^{41} - 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} - 18 q^{55} - 4 q^{57} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 12 q^{65} - 26 q^{67} + 24 q^{69} - 16 q^{71} + 22 q^{73} - 5 q^{75} - 12 q^{77} + 10 q^{79} + 3 q^{81} - 4 q^{83} - 24 q^{85} + 14 q^{87} - 6 q^{89} - 8 q^{91} + 4 q^{93} + 20 q^{95} + 6 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9 - 4 * q^11 + 6 * q^13 - 6 * q^15 + 4 * q^19 + 4 * q^21 - 24 * q^23 + 5 * q^25 - 3 * q^27 - 14 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 10 * q^37 - 6 * q^39 + 12 * q^41 - 8 * q^43 + 6 * q^45 + 11 * q^49 - 8 * q^53 - 18 * q^55 - 4 * q^57 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 12 * q^65 - 26 * q^67 + 24 * q^69 - 16 * q^71 + 22 * q^73 - 5 * q^75 - 12 * q^77 + 10 * q^79 + 3 * q^81 - 4 * q^83 - 24 * q^85 + 14 * q^87 - 6 * q^89 - 8 * q^91 + 4 * q^93 + 20 * q^95 + 6 * q^97 - 4 * 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
$5$	4.21432	1.88470	0.942351	−	0.334627i	$-0.108610\pi$
$5$	4.21432	1.88470	0.942351	+	0.334627i	$0.108610\pi$
$6$	0	0
$7$	−1.37778	−0.520754	−0.260377	−	0.965507i	$-0.583847\pi$
$7$	−1.37778	−0.520754	−0.260377	+	0.965507i	$0.583847\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.52543	−1.06296	−0.531478	−	0.847072i	$-0.678363\pi$
$11$	−3.52543	−1.06296	−0.531478	+	0.847072i	$0.678363\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−4.21432	−1.08813
$16$	0	0
$17$	−6.64296	−1.61115	−0.805577	−	0.592491i	$-0.798145\pi$
$17$	−6.64296	−1.61115	−0.805577	+	0.592491i	$0.798145\pi$
$18$	0	0
$19$	5.80642	1.33208	0.666042	−	0.745914i	$-0.267987\pi$
$19$	5.80642	1.33208	0.666042	+	0.745914i	$0.267987\pi$
$20$	0	0
$21$	1.37778	0.300657
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	12.7605	2.55210
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.05086	−1.68070	−0.840351	−	0.542043i	$-0.817651\pi$
$29$	−9.05086	−1.68070	−0.840351	+	0.542043i	$0.817651\pi$
$30$	0	0
$31$	−5.80642	−1.04286	−0.521432	−	0.853293i	$-0.674602\pi$
$31$	−5.80642	−1.04286	−0.521432	+	0.853293i	$0.674602\pi$
$32$	0	0
$33$	3.52543	0.613698
$34$	0	0
$35$	−5.80642	−0.981465
$36$	0	0
$37$	−3.37778	−0.555304	−0.277652	−	0.960682i	$-0.589556\pi$
$37$	−3.37778	−0.555304	−0.277652	+	0.960682i	$0.589556\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.7763	1.68298	0.841489	−	0.540275i	$-0.181680\pi$
$41$	10.7763	1.68298	0.841489	+	0.540275i	$0.181680\pi$
$42$	0	0
$43$	4.02074	0.613158	0.306579	−	0.951845i	$-0.400816\pi$
$43$	4.02074	0.613158	0.306579	+	0.951845i	$0.400816\pi$
$44$	0	0
$45$	4.21432	0.628234
$46$	0	0
$47$	−6.57628	−0.959249	−0.479625	−	0.877474i	$-0.659227\pi$
$47$	−6.57628	−0.959249	−0.479625	+	0.877474i	$0.659227\pi$
$48$	0	0
$49$	−5.10171	−0.728816
$50$	0	0
$51$	6.64296	0.930200
$52$	0	0
$53$	−9.39853	−1.29099	−0.645494	−	0.763766i	$-0.723348\pi$
$53$	−9.39853	−1.29099	−0.645494	+	0.763766i	$0.723348\pi$
$54$	0	0
$55$	−14.8573	−2.00336
$56$	0	0
$57$	−5.80642	−0.769080
$58$	0	0
$59$	−3.05086	−0.397188	−0.198594	−	0.980082i	$-0.563637\pi$
$59$	−3.05086	−0.397188	−0.198594	+	0.980082i	$0.563637\pi$
$60$	0	0
$61$	−10.5161	−1.34644	−0.673222	−	0.739441i	$-0.735090\pi$
$61$	−10.5161	−1.34644	−0.673222	+	0.739441i	$0.735090\pi$
$62$	0	0
$63$	−1.37778	−0.173585
$64$	0	0
$65$	8.42864	1.04544
$66$	0	0
$67$	−6.40790	−0.782849	−0.391425	−	0.920210i	$-0.628018\pi$
$67$	−6.40790	−0.782849	−0.391425	+	0.920210i	$0.628018\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	8.04149	0.954349	0.477174	−	0.878809i	$-0.341661\pi$
$71$	8.04149	0.954349	0.477174	+	0.878809i	$0.341661\pi$
$72$	0	0
$73$	16.2351	1.90017	0.950085	−	0.311991i	$-0.100996\pi$
$73$	16.2351	1.90017	0.950085	+	0.311991i	$0.100996\pi$
$74$	0	0
$75$	−12.7605	−1.47345
$76$	0	0
$77$	4.85728	0.553538
$78$	0	0
$79$	−7.69381	−0.865622	−0.432811	−	0.901485i	$-0.642478\pi$
$79$	−7.69381	−0.865622	−0.432811	+	0.901485i	$0.642478\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.7971	−1.62419	−0.812094	−	0.583527i	$-0.801672\pi$
$83$	−14.7971	−1.62419	−0.812094	+	0.583527i	$0.801672\pi$
$84$	0	0
$85$	−27.9956	−3.03654
$86$	0	0
$87$	9.05086	0.970354
$88$	0	0
$89$	−6.29529	−0.667299	−0.333650	−	0.942697i	$-0.608280\pi$
$89$	−6.29529	−0.667299	−0.333650	+	0.942697i	$0.608280\pi$
$90$	0	0
$91$	−2.75557	−0.288862
$92$	0	0
$93$	5.80642	0.602098
$94$	0	0
$95$	24.4701	2.51058
$96$	0	0
$97$	15.2859	1.55205	0.776025	−	0.630702i	$-0.217233\pi$
$97$	15.2859	1.55205	0.776025	+	0.630702i	$0.217233\pi$
$98$	0	0
$99$	−3.52543	−0.354319
$100$	0	0
$101$	−1.59210	−0.158420	−0.0792101	−	0.996858i	$-0.525240\pi$
$101$	−1.59210	−0.158420	−0.0792101	+	0.996858i	$0.525240\pi$
$102$	0	0
$103$	−9.59210	−0.945138	−0.472569	−	0.881294i	$-0.656673\pi$
$103$	−9.59210	−0.945138	−0.472569	+	0.881294i	$0.656673\pi$
$104$	0	0
$105$	5.80642	0.566649
$106$	0	0
$107$	−2.28100	−0.220512	−0.110256	−	0.993903i	$-0.535167\pi$
$107$	−2.28100	−0.220512	−0.110256	+	0.993903i	$0.535167\pi$
$108$	0	0
$109$	2.29529	0.219849	0.109924	−	0.993940i	$-0.464939\pi$
$109$	2.29529	0.219849	0.109924	+	0.993940i	$0.464939\pi$
$110$	0	0
$111$	3.37778	0.320605
$112$	0	0
$113$	−2.96989	−0.279384	−0.139692	−	0.990195i	$-0.544611\pi$
$113$	−2.96989	−0.279384	−0.139692	+	0.990195i	$0.544611\pi$
$114$	0	0
$115$	−33.7146	−3.14390
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	9.15257	0.839014
$120$	0	0
$121$	1.42864	0.129876
$122$	0	0
$123$	−10.7763	−0.971667
$124$	0	0
$125$	32.7052	2.92524
$126$	0	0
$127$	1.67307	0.148461	0.0742305	−	0.997241i	$-0.476350\pi$
$127$	1.67307	0.148461	0.0742305	+	0.997241i	$0.476350\pi$
$128$	0	0
$129$	−4.02074	−0.354007
$130$	0	0
$131$	7.47949	0.653486	0.326743	−	0.945113i	$-0.394049\pi$
$131$	7.47949	0.653486	0.326743	+	0.945113i	$0.394049\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−4.21432	−0.362711
$136$	0	0
$137$	5.47949	0.468145	0.234072	−	0.972219i	$-0.424795\pi$
$137$	5.47949	0.468145	0.234072	+	0.972219i	$0.424795\pi$
$138$	0	0
$139$	14.2143	1.20564	0.602821	−	0.797876i	$-0.294043\pi$
$139$	14.2143	1.20564	0.602821	+	0.797876i	$0.294043\pi$
$140$	0	0
$141$	6.57628	0.553823
$142$	0	0
$143$	−7.05086	−0.589622
$144$	0	0
$145$	−38.1432	−3.16762
$146$	0	0
$147$	5.10171	0.420782
$148$	0	0
$149$	17.3067	1.41782	0.708908	−	0.705300i	$-0.249188\pi$
$149$	17.3067	1.41782	0.708908	+	0.705300i	$0.249188\pi$
$150$	0	0
$151$	7.88739	0.641867	0.320933	−	0.947102i	$-0.396003\pi$
$151$	7.88739	0.641867	0.320933	+	0.947102i	$0.396003\pi$
$152$	0	0
$153$	−6.64296	−0.537051
$154$	0	0
$155$	−24.4701	−1.96549
$156$	0	0
$157$	−0.326929	−0.0260918	−0.0130459	−	0.999915i	$-0.504153\pi$
$157$	−0.326929	−0.0260918	−0.0130459	+	0.999915i	$0.504153\pi$
$158$	0	0
$159$	9.39853	0.745352
$160$	0	0
$161$	11.0223	0.868677
$162$	0	0
$163$	−14.2143	−1.11335	−0.556676	−	0.830730i	$-0.687923\pi$
$163$	−14.2143	−1.11335	−0.556676	+	0.830730i	$0.687923\pi$
$164$	0	0
$165$	14.8573	1.15664
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	5.80642	0.444028
$172$	0	0
$173$	2.38715	0.181492	0.0907459	−	0.995874i	$-0.471075\pi$
$173$	2.38715	0.181492	0.0907459	+	0.995874i	$0.471075\pi$
$174$	0	0
$175$	−17.5812	−1.32901
$176$	0	0
$177$	3.05086	0.229316
$178$	0	0
$179$	−3.14272	−0.234898	−0.117449	−	0.993079i	$-0.537472\pi$
$179$	−3.14272	−0.234898	−0.117449	+	0.993079i	$0.537472\pi$
$180$	0	0
$181$	17.0049	1.26397	0.631983	−	0.774982i	$-0.282241\pi$
$181$	17.0049	1.26397	0.631983	+	0.774982i	$0.282241\pi$
$182$	0	0
$183$	10.5161	0.777369
$184$	0	0
$185$	−14.2351	−1.04658
$186$	0	0
$187$	23.4193	1.71259
$188$	0	0
$189$	1.37778	0.100219
$190$	0	0
$191$	−6.66815	−0.482490	−0.241245	−	0.970464i	$-0.577556\pi$
$191$	−6.66815	−0.482490	−0.241245	+	0.970464i	$0.577556\pi$
$192$	0	0
$193$	9.05086	0.651495	0.325747	−	0.945457i	$-0.394384\pi$
$193$	9.05086	0.651495	0.325747	+	0.945457i	$0.394384\pi$
$194$	0	0
$195$	−8.42864	−0.603587
$196$	0	0
$197$	−22.8780	−1.62999	−0.814996	−	0.579467i	$-0.803261\pi$
$197$	−22.8780	−1.62999	−0.814996	+	0.579467i	$0.803261\pi$
$198$	0	0
$199$	12.0415	0.853598	0.426799	−	0.904346i	$-0.359641\pi$
$199$	12.0415	0.853598	0.426799	+	0.904346i	$0.359641\pi$
$200$	0	0
$201$	6.40790	0.451978
$202$	0	0
$203$	12.4701	0.875231
$204$	0	0
$205$	45.4148	3.17191
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	−20.4701	−1.41595
$210$	0	0
$211$	−7.34614	−0.505729	−0.252865	−	0.967502i	$-0.581373\pi$
$211$	−7.34614	−0.505729	−0.252865	+	0.967502i	$0.581373\pi$
$212$	0	0
$213$	−8.04149	−0.550994
$214$	0	0
$215$	16.9447	1.15562
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	−16.2351	−1.09706
$220$	0	0
$221$	−13.2859	−0.893708
$222$	0	0
$223$	−1.37778	−0.0922633	−0.0461316	−	0.998935i	$-0.514689\pi$
$223$	−1.37778	−0.0922633	−0.0461316	+	0.998935i	$0.514689\pi$
$224$	0	0
$225$	12.7605	0.850699
$226$	0	0
$227$	28.0830	1.86393	0.931966	−	0.362545i	$-0.118092\pi$
$227$	28.0830	1.86393	0.931966	+	0.362545i	$0.118092\pi$
$228$	0	0
$229$	−15.3733	−1.01590	−0.507949	−	0.861387i	$-0.669596\pi$
$229$	−15.3733	−1.01590	−0.507949	+	0.861387i	$0.669596\pi$
$230$	0	0
$231$	−4.85728	−0.319585
$232$	0	0
$233$	−22.7239	−1.48869	−0.744347	−	0.667793i	$-0.767239\pi$
$233$	−22.7239	−1.48869	−0.744347	+	0.667793i	$0.767239\pi$
$234$	0	0
$235$	−27.7146	−1.80790
$236$	0	0
$237$	7.69381	0.499767
$238$	0	0
$239$	5.06515	0.327637	0.163819	−	0.986490i	$-0.447619\pi$
$239$	5.06515	0.327637	0.163819	+	0.986490i	$0.447619\pi$
$240$	0	0
$241$	−17.7462	−1.14313	−0.571567	−	0.820556i	$-0.693664\pi$
$241$	−17.7462	−1.14313	−0.571567	+	0.820556i	$0.693664\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−21.5002	−1.37360
$246$	0	0
$247$	11.6128	0.738908
$248$	0	0
$249$	14.7971	0.937725
$250$	0	0
$251$	26.7052	1.68562	0.842808	−	0.538214i	$-0.180901\pi$
$251$	26.7052	1.68562	0.842808	+	0.538214i	$0.180901\pi$
$252$	0	0
$253$	28.2034	1.77313
$254$	0	0
$255$	27.9956	1.75315
$256$	0	0
$257$	−18.1541	−1.13242	−0.566211	−	0.824261i	$-0.691591\pi$
$257$	−18.1541	−1.13242	−0.566211	+	0.824261i	$0.691591\pi$
$258$	0	0
$259$	4.65386	0.289177
$260$	0	0
$261$	−9.05086	−0.560234
$262$	0	0
$263$	−25.4608	−1.56998	−0.784989	−	0.619510i	$-0.787331\pi$
$263$	−25.4608	−1.56998	−0.784989	+	0.619510i	$0.787331\pi$
$264$	0	0
$265$	−39.6084	−2.43312
$266$	0	0
$267$	6.29529	0.385265
$268$	0	0
$269$	−17.8272	−1.08694	−0.543471	−	0.839428i	$-0.682890\pi$
$269$	−17.8272	−1.08694	−0.543471	+	0.839428i	$0.682890\pi$
$270$	0	0
$271$	−7.39853	−0.449429	−0.224714	−	0.974425i	$-0.572145\pi$
$271$	−7.39853	−0.449429	−0.224714	+	0.974425i	$0.572145\pi$
$272$	0	0
$273$	2.75557	0.166775
$274$	0	0
$275$	−44.9862	−2.71277
$276$	0	0
$277$	−25.2543	−1.51738	−0.758691	−	0.651450i	$-0.774161\pi$
$277$	−25.2543	−1.51738	−0.758691	+	0.651450i	$0.774161\pi$
$278$	0	0
$279$	−5.80642	−0.347622
$280$	0	0
$281$	−10.7239	−0.639736	−0.319868	−	0.947462i	$-0.603639\pi$
$281$	−10.7239	−0.639736	−0.319868	+	0.947462i	$0.603639\pi$
$282$	0	0
$283$	4.09187	0.243236	0.121618	−	0.992577i	$-0.461192\pi$
$283$	4.09187	0.243236	0.121618	+	0.992577i	$0.461192\pi$
$284$	0	0
$285$	−24.4701	−1.44949
$286$	0	0
$287$	−14.8474	−0.876416
$288$	0	0
$289$	27.1289	1.59582
$290$	0	0
$291$	−15.2859	−0.896076
$292$	0	0
$293$	−18.7368	−1.09462	−0.547309	−	0.836931i	$-0.684348\pi$
$293$	−18.7368	−1.09462	−0.547309	+	0.836931i	$0.684348\pi$
$294$	0	0
$295$	−12.8573	−0.748580
$296$	0	0
$297$	3.52543	0.204566
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−5.53972	−0.319304
$302$	0	0
$303$	1.59210	0.0914640
$304$	0	0
$305$	−44.3180	−2.53764
$306$	0	0
$307$	0.969888	0.0553545	0.0276772	−	0.999617i	$-0.491189\pi$
$307$	0.969888	0.0553545	0.0276772	+	0.999617i	$0.491189\pi$
$308$	0	0
$309$	9.59210	0.545676
$310$	0	0
$311$	15.7462	0.892885	0.446443	−	0.894812i	$-0.352691\pi$
$311$	15.7462	0.892885	0.446443	+	0.894812i	$0.352691\pi$
$312$	0	0
$313$	5.61285	0.317257	0.158628	−	0.987338i	$-0.449293\pi$
$313$	5.61285	0.317257	0.158628	+	0.987338i	$0.449293\pi$
$314$	0	0
$315$	−5.80642	−0.327155
$316$	0	0
$317$	10.5620	0.593221	0.296610	−	0.954999i	$-0.404144\pi$
$317$	10.5620	0.593221	0.296610	+	0.954999i	$0.404144\pi$
$318$	0	0
$319$	31.9081	1.78651
$320$	0	0
$321$	2.28100	0.127313
$322$	0	0
$323$	−38.5718	−2.14619
$324$	0	0
$325$	25.5210	1.41565
$326$	0	0
$327$	−2.29529	−0.126930
$328$	0	0
$329$	9.06070	0.499533
$330$	0	0
$331$	−19.0400	−1.04653	−0.523265	−	0.852170i	$-0.675286\pi$
$331$	−19.0400	−1.04653	−0.523265	+	0.852170i	$0.675286\pi$
$332$	0	0
$333$	−3.37778	−0.185101
$334$	0	0
$335$	−27.0049	−1.47544
$336$	0	0
$337$	4.79706	0.261312	0.130656	−	0.991428i	$-0.458292\pi$
$337$	4.79706	0.261312	0.130656	+	0.991428i	$0.458292\pi$
$338$	0	0
$339$	2.96989	0.161302
$340$	0	0
$341$	20.4701	1.10852
$342$	0	0
$343$	16.6735	0.900287
$344$	0	0
$345$	33.7146	1.81513
$346$	0	0
$347$	−14.7841	−0.793655	−0.396827	−	0.917893i	$-0.629889\pi$
$347$	−14.7841	−0.793655	−0.396827	+	0.917893i	$0.629889\pi$
$348$	0	0
$349$	−2.52051	−0.134920	−0.0674598	−	0.997722i	$-0.521489\pi$
$349$	−2.52051	−0.134920	−0.0674598	+	0.997722i	$0.521489\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	21.4795	1.14324	0.571619	−	0.820519i	$-0.306316\pi$
$353$	21.4795	1.14324	0.571619	+	0.820519i	$0.306316\pi$
$354$	0	0
$355$	33.8894	1.79866
$356$	0	0
$357$	−9.15257	−0.484405
$358$	0	0
$359$	3.34614	0.176603	0.0883013	−	0.996094i	$-0.471856\pi$
$359$	3.34614	0.176603	0.0883013	+	0.996094i	$0.471856\pi$
$360$	0	0
$361$	14.7146	0.774450
$362$	0	0
$363$	−1.42864	−0.0749841
$364$	0	0
$365$	68.4197	3.58125
$366$	0	0
$367$	2.07313	0.108217	0.0541083	−	0.998535i	$-0.482768\pi$
$367$	2.07313	0.108217	0.0541083	+	0.998535i	$0.482768\pi$
$368$	0	0
$369$	10.7763	0.560992
$370$	0	0
$371$	12.9491	0.672286
$372$	0	0
$373$	−28.6133	−1.48154	−0.740771	−	0.671758i	$-0.765540\pi$
$373$	−28.6133	−1.48154	−0.740771	+	0.671758i	$0.765540\pi$
$374$	0	0
$375$	−32.7052	−1.68889
$376$	0	0
$377$	−18.1017	−0.932286
$378$	0	0
$379$	30.5926	1.57144	0.785718	−	0.618585i	$-0.212294\pi$
$379$	30.5926	1.57144	0.785718	+	0.618585i	$0.212294\pi$
$380$	0	0
$381$	−1.67307	−0.0857140
$382$	0	0
$383$	1.03657	0.0529660	0.0264830	−	0.999649i	$-0.491569\pi$
$383$	1.03657	0.0529660	0.0264830	+	0.999649i	$0.491569\pi$
$384$	0	0
$385$	20.4701	1.04325
$386$	0	0
$387$	4.02074	0.204386
$388$	0	0
$389$	−5.12198	−0.259695	−0.129847	−	0.991534i	$-0.541449\pi$
$389$	−5.12198	−0.259695	−0.129847	+	0.991534i	$0.541449\pi$
$390$	0	0
$391$	53.1437	2.68759
$392$	0	0
$393$	−7.47949	−0.377291
$394$	0	0
$395$	−32.4242	−1.63144
$396$	0	0
$397$	−13.8336	−0.694290	−0.347145	−	0.937812i	$-0.612849\pi$
$397$	−13.8336	−0.694290	−0.347145	+	0.937812i	$0.612849\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−10.4079	−0.519746	−0.259873	−	0.965643i	$-0.583681\pi$
$401$	−10.4079	−0.519746	−0.259873	+	0.965643i	$0.583681\pi$
$402$	0	0
$403$	−11.6128	−0.578477
$404$	0	0
$405$	4.21432	0.209411
$406$	0	0
$407$	11.9081	0.590264
$408$	0	0
$409$	30.8113	1.52352	0.761762	−	0.647858i	$-0.224335\pi$
$409$	30.8113	1.52352	0.761762	+	0.647858i	$0.224335\pi$
$410$	0	0
$411$	−5.47949	−0.270284
$412$	0	0
$413$	4.20342	0.206837
$414$	0	0
$415$	−62.3595	−3.06111
$416$	0	0
$417$	−14.2143	−0.696078
$418$	0	0
$419$	10.8716	0.531111	0.265555	−	0.964096i	$-0.414445\pi$
$419$	10.8716	0.531111	0.265555	+	0.964096i	$0.414445\pi$
$420$	0	0
$421$	−26.8988	−1.31097	−0.655483	−	0.755210i	$-0.727535\pi$
$421$	−26.8988	−1.31097	−0.655483	+	0.755210i	$0.727535\pi$
$422$	0	0
$423$	−6.57628	−0.319750
$424$	0	0
$425$	−84.7674	−4.11182
$426$	0	0
$427$	14.4889	0.701165
$428$	0	0
$429$	7.05086	0.340418
$430$	0	0
$431$	−37.5812	−1.81022	−0.905111	−	0.425174i	$-0.860213\pi$
$431$	−37.5812	−1.81022	−0.905111	+	0.425174i	$0.860213\pi$
$432$	0	0
$433$	−1.68736	−0.0810894	−0.0405447	−	0.999178i	$-0.512909\pi$
$433$	−1.68736	−0.0810894	−0.0405447	+	0.999178i	$0.512909\pi$
$434$	0	0
$435$	38.1432	1.82883
$436$	0	0
$437$	−46.4514	−2.22207
$438$	0	0
$439$	−18.9512	−0.904489	−0.452245	−	0.891894i	$-0.649377\pi$
$439$	−18.9512	−0.904489	−0.452245	+	0.891894i	$0.649377\pi$
$440$	0	0
$441$	−5.10171	−0.242939
$442$	0	0
$443$	−24.7654	−1.17664	−0.588320	−	0.808628i	$-0.700211\pi$
$443$	−24.7654	−1.17664	−0.588320	+	0.808628i	$0.700211\pi$
$444$	0	0
$445$	−26.5303	−1.25766
$446$	0	0
$447$	−17.3067	−0.818577
$448$	0	0
$449$	−26.3082	−1.24156	−0.620780	−	0.783985i	$-0.713184\pi$
$449$	−26.3082	−1.24156	−0.620780	+	0.783985i	$0.713184\pi$
$450$	0	0
$451$	−37.9911	−1.78893
$452$	0	0
$453$	−7.88739	−0.370582
$454$	0	0
$455$	−11.6128	−0.544419
$456$	0	0
$457$	−10.0415	−0.469721	−0.234860	−	0.972029i	$-0.575463\pi$
$457$	−10.0415	−0.469721	−0.234860	+	0.972029i	$0.575463\pi$
$458$	0	0
$459$	6.64296	0.310067
$460$	0	0
$461$	34.4286	1.60350	0.801751	−	0.597658i	$-0.203902\pi$
$461$	34.4286	1.60350	0.801751	+	0.597658i	$0.203902\pi$
$462$	0	0
$463$	31.9605	1.48533	0.742666	−	0.669662i	$-0.233561\pi$
$463$	31.9605	1.48533	0.742666	+	0.669662i	$0.233561\pi$
$464$	0	0
$465$	24.4701	1.13477
$466$	0	0
$467$	30.3051	1.40235	0.701177	−	0.712987i	$-0.252658\pi$
$467$	30.3051	1.40235	0.701177	+	0.712987i	$0.252658\pi$
$468$	0	0
$469$	8.82870	0.407671
$470$	0	0
$471$	0.326929	0.0150641
$472$	0	0
$473$	−14.1748	−0.651760
$474$	0	0
$475$	74.0928	3.39961
$476$	0	0
$477$	−9.39853	−0.430329
$478$	0	0
$479$	−9.28592	−0.424284	−0.212142	−	0.977239i	$-0.568044\pi$
$479$	−9.28592	−0.424284	−0.212142	+	0.977239i	$0.568044\pi$
$480$	0	0
$481$	−6.75557	−0.308027
$482$	0	0
$483$	−11.0223	−0.501531
$484$	0	0
$485$	64.4197	2.92515
$486$	0	0
$487$	−43.4400	−1.96845	−0.984227	−	0.176907i	$-0.943391\pi$
$487$	−43.4400	−1.96845	−0.984227	+	0.176907i	$0.943391\pi$
$488$	0	0
$489$	14.2143	0.642794
$490$	0	0
$491$	0.649413	0.0293076	0.0146538	−	0.999893i	$-0.495335\pi$
$491$	0.649413	0.0293076	0.0146538	+	0.999893i	$0.495335\pi$
$492$	0	0
$493$	60.1245	2.70787
$494$	0	0
$495$	−14.8573	−0.667785
$496$	0	0
$497$	−11.0794	−0.496981
$498$	0	0
$499$	−5.63359	−0.252194	−0.126097	−	0.992018i	$-0.540245\pi$
$499$	−5.63359	−0.252194	−0.126097	+	0.992018i	$0.540245\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−3.05530	−0.136229	−0.0681146	−	0.997678i	$-0.521698\pi$
$503$	−3.05530	−0.136229	−0.0681146	+	0.997678i	$0.521698\pi$
$504$	0	0
$505$	−6.70964	−0.298575
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−3.40636	−0.150984	−0.0754922	−	0.997146i	$-0.524053\pi$
$509$	−3.40636	−0.150984	−0.0754922	+	0.997146i	$0.524053\pi$
$510$	0	0
$511$	−22.3684	−0.989520
$512$	0	0
$513$	−5.80642	−0.256360
$514$	0	0
$515$	−40.4242	−1.78130
$516$	0	0
$517$	23.1842	1.01964
$518$	0	0
$519$	−2.38715	−0.104784
$520$	0	0
$521$	19.2335	0.842636	0.421318	−	0.906913i	$-0.361568\pi$
$521$	19.2335	0.842636	0.421318	+	0.906913i	$0.361568\pi$
$522$	0	0
$523$	−42.8385	−1.87320	−0.936599	−	0.350402i	$-0.886045\pi$
$523$	−42.8385	−1.87320	−0.936599	+	0.350402i	$0.886045\pi$
$524$	0	0
$525$	17.5812	0.767307
$526$	0	0
$527$	38.5718	1.68022
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−3.05086	−0.132396
$532$	0	0
$533$	21.5526	0.933548
$534$	0	0
$535$	−9.61285	−0.415600
$536$	0	0
$537$	3.14272	0.135618
$538$	0	0
$539$	17.9857	0.774699
$540$	0	0
$541$	−33.4291	−1.43723	−0.718615	−	0.695408i	$-0.755224\pi$
$541$	−33.4291	−1.43723	−0.718615	+	0.695408i	$0.755224\pi$
$542$	0	0
$543$	−17.0049	−0.729751
$544$	0	0
$545$	9.67307	0.414349
$546$	0	0
$547$	34.7447	1.48557	0.742787	−	0.669527i	$-0.233503\pi$
$547$	34.7447	1.48557	0.742787	+	0.669527i	$0.233503\pi$
$548$	0	0
$549$	−10.5161	−0.448814
$550$	0	0
$551$	−52.5531	−2.23884
$552$	0	0
$553$	10.6004	0.450776
$554$	0	0
$555$	14.2351	0.604245
$556$	0	0
$557$	11.8479	0.502012	0.251006	−	0.967986i	$-0.419239\pi$
$557$	11.8479	0.502012	0.251006	+	0.967986i	$0.419239\pi$
$558$	0	0
$559$	8.04149	0.340119
$560$	0	0
$561$	−23.4193	−0.988762
$562$	0	0
$563$	21.9684	0.925856	0.462928	−	0.886396i	$-0.346799\pi$
$563$	21.9684	0.925856	0.462928	+	0.886396i	$0.346799\pi$
$564$	0	0
$565$	−12.5161	−0.526555
$566$	0	0
$567$	−1.37778	−0.0578615
$568$	0	0
$569$	25.5131	1.06957	0.534783	−	0.844989i	$-0.320393\pi$
$569$	25.5131	1.06957	0.534783	+	0.844989i	$0.320393\pi$
$570$	0	0
$571$	26.7259	1.11845	0.559223	−	0.829017i	$-0.311100\pi$
$571$	26.7259	1.11845	0.559223	+	0.829017i	$0.311100\pi$
$572$	0	0
$573$	6.66815	0.278566
$574$	0	0
$575$	−102.084	−4.25719
$576$	0	0
$577$	−38.3037	−1.59461	−0.797303	−	0.603579i	$-0.793741\pi$
$577$	−38.3037	−1.59461	−0.797303	+	0.603579i	$0.793741\pi$
$578$	0	0
$579$	−9.05086	−0.376141
$580$	0	0
$581$	20.3872	0.845802
$582$	0	0
$583$	33.1338	1.37226
$584$	0	0
$585$	8.42864	0.348481
$586$	0	0
$587$	20.9403	0.864297	0.432148	−	0.901803i	$-0.357756\pi$
$587$	20.9403	0.864297	0.432148	+	0.901803i	$0.357756\pi$
$588$	0	0
$589$	−33.7146	−1.38918
$590$	0	0
$591$	22.8780	0.941076
$592$	0	0
$593$	−18.4909	−0.759329	−0.379665	−	0.925124i	$-0.623961\pi$
$593$	−18.4909	−0.759329	−0.379665	+	0.925124i	$0.623961\pi$
$594$	0	0
$595$	38.5718	1.58129
$596$	0	0
$597$	−12.0415	−0.492825
$598$	0	0
$599$	0.590573	0.0241301	0.0120651	−	0.999927i	$-0.496159\pi$
$599$	0.590573	0.0241301	0.0120651	+	0.999927i	$0.496159\pi$
$600$	0	0
$601$	−27.3274	−1.11471	−0.557354	−	0.830275i	$-0.688183\pi$
$601$	−27.3274	−1.11471	−0.557354	+	0.830275i	$0.688183\pi$
$602$	0	0
$603$	−6.40790	−0.260950
$604$	0	0
$605$	6.02074	0.244778
$606$	0	0
$607$	33.2652	1.35019	0.675096	−	0.737730i	$-0.264102\pi$
$607$	33.2652	1.35019	0.675096	+	0.737730i	$0.264102\pi$
$608$	0	0
$609$	−12.4701	−0.505315
$610$	0	0
$611$	−13.1526	−0.532096
$612$	0	0
$613$	0.147643	0.00596325	0.00298163	−	0.999996i	$-0.499051\pi$
$613$	0.147643	0.00596325	0.00298163	+	0.999996i	$0.499051\pi$
$614$	0	0
$615$	−45.4148	−1.83130
$616$	0	0
$617$	0.755569	0.0304181	0.0152090	−	0.999884i	$-0.495159\pi$
$617$	0.755569	0.0304181	0.0152090	+	0.999884i	$0.495159\pi$
$618$	0	0
$619$	33.1635	1.33295	0.666476	−	0.745526i	$-0.267802\pi$
$619$	33.1635	1.33295	0.666476	+	0.745526i	$0.267802\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	8.67355	0.347498
$624$	0	0
$625$	74.0277	2.96111
$626$	0	0
$627$	20.4701	0.817498
$628$	0	0
$629$	22.4385	0.894681
$630$	0	0
$631$	16.1619	0.643396	0.321698	−	0.946842i	$-0.395746\pi$
$631$	16.1619	0.643396	0.321698	+	0.946842i	$0.395746\pi$
$632$	0	0
$633$	7.34614	0.291983
$634$	0	0
$635$	7.05086	0.279805
$636$	0	0
$637$	−10.2034	−0.404274
$638$	0	0
$639$	8.04149	0.318116
$640$	0	0
$641$	−3.74419	−0.147887	−0.0739434	−	0.997262i	$-0.523558\pi$
$641$	−3.74419	−0.147887	−0.0739434	+	0.997262i	$0.523558\pi$
$642$	0	0
$643$	24.1510	0.952424	0.476212	−	0.879331i	$-0.342010\pi$
$643$	24.1510	0.952424	0.476212	+	0.879331i	$0.342010\pi$
$644$	0	0
$645$	−16.9447	−0.667197
$646$	0	0
$647$	4.96205	0.195078	0.0975392	−	0.995232i	$-0.468903\pi$
$647$	4.96205	0.195078	0.0975392	+	0.995232i	$0.468903\pi$
$648$	0	0
$649$	10.7556	0.422193
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	22.4701	0.879324	0.439662	−	0.898163i	$-0.355098\pi$
$653$	22.4701	0.879324	0.439662	+	0.898163i	$0.355098\pi$
$654$	0	0
$655$	31.5210	1.23163
$656$	0	0
$657$	16.2351	0.633390
$658$	0	0
$659$	−33.2988	−1.29714	−0.648569	−	0.761156i	$-0.724632\pi$
$659$	−33.2988	−1.29714	−0.648569	+	0.761156i	$0.724632\pi$
$660$	0	0
$661$	25.1753	0.979206	0.489603	−	0.871945i	$-0.337142\pi$
$661$	25.1753	0.979206	0.489603	+	0.871945i	$0.337142\pi$
$662$	0	0
$663$	13.2859	0.515982
$664$	0	0
$665$	−33.7146	−1.30739
$666$	0	0
$667$	72.4068	2.80360
$668$	0	0
$669$	1.37778	0.0532682
$670$	0	0
$671$	37.0736	1.43121
$672$	0	0
$673$	41.8163	1.61190	0.805949	−	0.591985i	$-0.201655\pi$
$673$	41.8163	1.61190	0.805949	+	0.591985i	$0.201655\pi$
$674$	0	0
$675$	−12.7605	−0.491152
$676$	0	0
$677$	21.4924	0.826020	0.413010	−	0.910726i	$-0.364477\pi$
$677$	21.4924	0.826020	0.413010	+	0.910726i	$0.364477\pi$
$678$	0	0
$679$	−21.0607	−0.808235
$680$	0	0
$681$	−28.0830	−1.07614
$682$	0	0
$683$	−1.93978	−0.0742235	−0.0371118	−	0.999311i	$-0.511816\pi$
$683$	−1.93978	−0.0742235	−0.0371118	+	0.999311i	$0.511816\pi$
$684$	0	0
$685$	23.0923	0.882313
$686$	0	0
$687$	15.3733	0.586529
$688$	0	0
$689$	−18.7971	−0.716111
$690$	0	0
$691$	43.0114	1.63623	0.818115	−	0.575055i	$-0.195019\pi$
$691$	43.0114	1.63623	0.818115	+	0.575055i	$0.195019\pi$
$692$	0	0
$693$	4.85728	0.184513
$694$	0	0
$695$	59.9037	2.27228
$696$	0	0
$697$	−71.5866	−2.71154
$698$	0	0
$699$	22.7239	0.859498
$700$	0	0
$701$	31.0192	1.17158	0.585790	−	0.810463i	$-0.300784\pi$
$701$	31.0192	1.17158	0.585790	+	0.810463i	$0.300784\pi$
$702$	0	0
$703$	−19.6128	−0.739713
$704$	0	0
$705$	27.7146	1.04379
$706$	0	0
$707$	2.19358	0.0824979
$708$	0	0
$709$	−11.1240	−0.417770	−0.208885	−	0.977940i	$-0.566983\pi$
$709$	−11.1240	−0.417770	−0.208885	+	0.977940i	$0.566983\pi$
$710$	0	0
$711$	−7.69381	−0.288541
$712$	0	0
$713$	46.4514	1.73962
$714$	0	0
$715$	−29.7146	−1.11126
$716$	0	0
$717$	−5.06515	−0.189161
$718$	0	0
$719$	−6.14320	−0.229103	−0.114551	−	0.993417i	$-0.536543\pi$
$719$	−6.14320	−0.229103	−0.114551	+	0.993417i	$0.536543\pi$
$720$	0	0
$721$	13.2159	0.492184
$722$	0	0
$723$	17.7462	0.659988
$724$	0	0
$725$	−115.493	−4.28932
$726$	0	0
$727$	18.6015	0.689890	0.344945	−	0.938623i	$-0.387897\pi$
$727$	18.6015	0.689890	0.344945	+	0.938623i	$0.387897\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.7096	−0.987892
$732$	0	0
$733$	−34.4528	−1.27254	−0.636271	−	0.771466i	$-0.719524\pi$
$733$	−34.4528	−1.27254	−0.636271	+	0.771466i	$0.719524\pi$
$734$	0	0
$735$	21.5002	0.793048
$736$	0	0
$737$	22.5906	0.832134
$738$	0	0
$739$	−16.9512	−0.623558	−0.311779	−	0.950155i	$-0.600925\pi$
$739$	−16.9512	−0.623558	−0.311779	+	0.950155i	$0.600925\pi$
$740$	0	0
$741$	−11.6128	−0.426609
$742$	0	0
$743$	9.98126	0.366177	0.183089	−	0.983096i	$-0.441390\pi$
$743$	9.98126	0.366177	0.183089	+	0.983096i	$0.441390\pi$
$744$	0	0
$745$	72.9358	2.67216
$746$	0	0
$747$	−14.7971	−0.541396
$748$	0	0
$749$	3.14272	0.114833
$750$	0	0
$751$	42.3477	1.54529	0.772644	−	0.634839i	$-0.218934\pi$
$751$	42.3477	1.54529	0.772644	+	0.634839i	$0.218934\pi$
$752$	0	0
$753$	−26.7052	−0.973191
$754$	0	0
$755$	33.2400	1.20973
$756$	0	0
$757$	43.9180	1.59623	0.798113	−	0.602508i	$-0.205832\pi$
$757$	43.9180	1.59623	0.798113	+	0.602508i	$0.205832\pi$
$758$	0	0
$759$	−28.2034	−1.02372
$760$	0	0
$761$	25.2257	0.914431	0.457215	−	0.889356i	$-0.348847\pi$
$761$	25.2257	0.914431	0.457215	+	0.889356i	$0.348847\pi$
$762$	0	0
$763$	−3.16241	−0.114487
$764$	0	0
$765$	−27.9956	−1.01218
$766$	0	0
$767$	−6.10171	−0.220320
$768$	0	0
$769$	23.9813	0.864787	0.432393	−	0.901685i	$-0.357669\pi$
$769$	23.9813	0.864787	0.432393	+	0.901685i	$0.357669\pi$
$770$	0	0
$771$	18.1541	0.653804
$772$	0	0
$773$	33.2050	1.19430	0.597150	−	0.802130i	$-0.296300\pi$
$773$	33.2050	1.19430	0.597150	+	0.802130i	$0.296300\pi$
$774$	0	0
$775$	−74.0928	−2.66149
$776$	0	0
$777$	−4.65386	−0.166956
$778$	0	0
$779$	62.5718	2.24187
$780$	0	0
$781$	−28.3497	−1.01443
$782$	0	0
$783$	9.05086	0.323451
$784$	0	0
$785$	−1.37778	−0.0491752
$786$	0	0
$787$	3.45875	0.123291	0.0616456	−	0.998098i	$-0.480365\pi$
$787$	3.45875	0.123291	0.0616456	+	0.998098i	$0.480365\pi$
$788$	0	0
$789$	25.4608	0.906427
$790$	0	0
$791$	4.09187	0.145490
$792$	0	0
$793$	−21.0321	−0.746872
$794$	0	0
$795$	39.6084	1.40477
$796$	0	0
$797$	20.5412	0.727608	0.363804	−	0.931475i	$-0.381478\pi$
$797$	20.5412	0.727608	0.363804	+	0.931475i	$0.381478\pi$
$798$	0	0
$799$	43.6860	1.54550
$800$	0	0
$801$	−6.29529	−0.222433
$802$	0	0
$803$	−57.2355	−2.01980
$804$	0	0
$805$	46.4514	1.63720
$806$	0	0
$807$	17.8272	0.627546
$808$	0	0
$809$	−34.4197	−1.21013	−0.605067	−	0.796175i	$-0.706854\pi$
$809$	−34.4197	−1.21013	−0.605067	+	0.796175i	$0.706854\pi$
$810$	0	0
$811$	38.8178	1.36308	0.681539	−	0.731782i	$-0.261311\pi$
$811$	38.8178	1.36308	0.681539	+	0.731782i	$0.261311\pi$
$812$	0	0
$813$	7.39853	0.259478
$814$	0	0
$815$	−59.9037	−2.09833
$816$	0	0
$817$	23.3461	0.816778
$818$	0	0
$819$	−2.75557	−0.0962874
$820$	0	0
$821$	2.37625	0.0829318	0.0414659	−	0.999140i	$-0.486797\pi$
$821$	2.37625	0.0829318	0.0414659	+	0.999140i	$0.486797\pi$
$822$	0	0
$823$	−4.21432	−0.146902	−0.0734510	−	0.997299i	$-0.523401\pi$
$823$	−4.21432	−0.146902	−0.0734510	+	0.997299i	$0.523401\pi$
$824$	0	0
$825$	44.9862	1.56622
$826$	0	0
$827$	−28.2034	−0.980729	−0.490365	−	0.871517i	$-0.663136\pi$
$827$	−28.2034	−0.980729	−0.490365	+	0.871517i	$0.663136\pi$
$828$	0	0
$829$	−26.2953	−0.913273	−0.456637	−	0.889653i	$-0.650946\pi$
$829$	−26.2953	−0.913273	−0.456637	+	0.889653i	$0.650946\pi$
$830$	0	0
$831$	25.2543	0.876061
$832$	0	0
$833$	33.8905	1.17423
$834$	0	0
$835$	4.21432	0.145843
$836$	0	0
$837$	5.80642	0.200699
$838$	0	0
$839$	−36.9733	−1.27646	−0.638230	−	0.769846i	$-0.720333\pi$
$839$	−36.9733	−1.27646	−0.638230	+	0.769846i	$0.720333\pi$
$840$	0	0
$841$	52.9180	1.82476
$842$	0	0
$843$	10.7239	0.369352
$844$	0	0
$845$	−37.9289	−1.30479
$846$	0	0
$847$	−1.96836	−0.0676336
$848$	0	0
$849$	−4.09187	−0.140432
$850$	0	0
$851$	27.0223	0.926312
$852$	0	0
$853$	10.5161	0.360063	0.180032	−	0.983661i	$-0.442380\pi$
$853$	10.5161	0.360063	0.180032	+	0.983661i	$0.442380\pi$
$854$	0	0
$855$	24.4701	0.836861
$856$	0	0
$857$	−4.10171	−0.140112	−0.0700559	−	0.997543i	$-0.522318\pi$
$857$	−4.10171	−0.140112	−0.0700559	+	0.997543i	$0.522318\pi$
$858$	0	0
$859$	−7.18421	−0.245122	−0.122561	−	0.992461i	$-0.539111\pi$
$859$	−7.18421	−0.245122	−0.122561	+	0.992461i	$0.539111\pi$
$860$	0	0
$861$	14.8474	0.505999
$862$	0	0
$863$	18.0513	0.614474	0.307237	−	0.951633i	$-0.400595\pi$
$863$	18.0513	0.614474	0.307237	+	0.951633i	$0.400595\pi$
$864$	0	0
$865$	10.0602	0.342058
$866$	0	0
$867$	−27.1289	−0.921346
$868$	0	0
$869$	27.1240	0.920118
$870$	0	0
$871$	−12.8158	−0.434247
$872$	0	0
$873$	15.2859	0.517350
$874$	0	0
$875$	−45.0607	−1.52333
$876$	0	0
$877$	8.20648	0.277113	0.138557	−	0.990355i	$-0.455754\pi$
$877$	8.20648	0.277113	0.138557	+	0.990355i	$0.455754\pi$
$878$	0	0
$879$	18.7368	0.631978
$880$	0	0
$881$	19.6938	0.663501	0.331751	−	0.943367i	$-0.392361\pi$
$881$	19.6938	0.663501	0.331751	+	0.943367i	$0.392361\pi$
$882$	0	0
$883$	−31.7244	−1.06761	−0.533806	−	0.845607i	$-0.679239\pi$
$883$	−31.7244	−1.06761	−0.533806	+	0.845607i	$0.679239\pi$
$884$	0	0
$885$	12.8573	0.432193
$886$	0	0
$887$	40.9273	1.37421	0.687103	−	0.726560i	$-0.258882\pi$
$887$	40.9273	1.37421	0.687103	+	0.726560i	$0.258882\pi$
$888$	0	0
$889$	−2.30513	−0.0773116
$890$	0	0
$891$	−3.52543	−0.118106
$892$	0	0
$893$	−38.1847	−1.27780
$894$	0	0
$895$	−13.2444	−0.442713
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	52.5531	1.75274
$900$	0	0
$901$	62.4340	2.07998
$902$	0	0
$903$	5.53972	0.184350
$904$	0	0
$905$	71.6642	2.38220
$906$	0	0
$907$	−41.5526	−1.37973	−0.689866	−	0.723937i	$-0.742331\pi$
$907$	−41.5526	−1.37973	−0.689866	+	0.723937i	$0.742331\pi$
$908$	0	0
$909$	−1.59210	−0.0528068
$910$	0	0
$911$	−37.1437	−1.23062	−0.615312	−	0.788283i	$-0.710970\pi$
$911$	−37.1437	−1.23062	−0.615312	+	0.788283i	$0.710970\pi$
$912$	0	0
$913$	52.1659	1.72644
$914$	0	0
$915$	44.3180	1.46511
$916$	0	0
$917$	−10.3051	−0.340305
$918$	0	0
$919$	−38.6766	−1.27582	−0.637912	−	0.770109i	$-0.720202\pi$
$919$	−38.6766	−1.27582	−0.637912	+	0.770109i	$0.720202\pi$
$920$	0	0
$921$	−0.969888	−0.0319589
$922$	0	0
$923$	16.0830	0.529378
$924$	0	0
$925$	−43.1022	−1.41719
$926$	0	0
$927$	−9.59210	−0.315046
$928$	0	0
$929$	15.9180	0.522252	0.261126	−	0.965305i	$-0.415906\pi$
$929$	15.9180	0.522252	0.261126	+	0.965305i	$0.415906\pi$
$930$	0	0
$931$	−29.6227	−0.970845
$932$	0	0
$933$	−15.7462	−0.515507
$934$	0	0
$935$	98.6963	3.22771
$936$	0	0
$937$	−41.5210	−1.35643	−0.678216	−	0.734863i	$-0.737246\pi$
$937$	−41.5210	−1.35643	−0.678216	+	0.734863i	$0.737246\pi$
$938$	0	0
$939$	−5.61285	−0.183168
$940$	0	0
$941$	−23.7768	−0.775101	−0.387551	−	0.921848i	$-0.626679\pi$
$941$	−23.7768	−0.775101	−0.387551	+	0.921848i	$0.626679\pi$
$942$	0	0
$943$	−86.2105	−2.80740
$944$	0	0
$945$	5.80642	0.188883
$946$	0	0
$947$	−49.8020	−1.61835	−0.809173	−	0.587570i	$-0.800085\pi$
$947$	−49.8020	−1.61835	−0.809173	+	0.587570i	$0.800085\pi$
$948$	0	0
$949$	32.4701	1.05402
$950$	0	0
$951$	−10.5620	−0.342496
$952$	0	0
$953$	−37.5002	−1.21475	−0.607376	−	0.794415i	$-0.707778\pi$
$953$	−37.5002	−1.21475	−0.607376	+	0.794415i	$0.707778\pi$
$954$	0	0
$955$	−28.1017	−0.909350
$956$	0	0
$957$	−31.9081	−1.03144
$958$	0	0
$959$	−7.54956	−0.243788
$960$	0	0
$961$	2.71456	0.0875664
$962$	0	0
$963$	−2.28100	−0.0735041
$964$	0	0
$965$	38.1432	1.22787
$966$	0	0
$967$	2.65080	0.0852438	0.0426219	−	0.999091i	$-0.486429\pi$
$967$	2.65080	0.0852438	0.0426219	+	0.999091i	$0.486429\pi$
$968$	0	0
$969$	38.5718	1.23911
$970$	0	0
$971$	−24.4197	−0.783667	−0.391834	−	0.920036i	$-0.628159\pi$
$971$	−24.4197	−0.783667	−0.391834	+	0.920036i	$0.628159\pi$
$972$	0	0
$973$	−19.5843	−0.627843
$974$	0	0
$975$	−25.5210	−0.817326
$976$	0	0
$977$	43.0212	1.37637	0.688185	−	0.725535i	$-0.258408\pi$
$977$	43.0212	1.37637	0.688185	+	0.725535i	$0.258408\pi$
$978$	0	0
$979$	22.1936	0.709310
$980$	0	0
$981$	2.29529	0.0732829
$982$	0	0
$983$	17.9684	0.573102	0.286551	−	0.958065i	$-0.407491\pi$
$983$	17.9684	0.573102	0.286551	+	0.958065i	$0.407491\pi$
$984$	0	0
$985$	−96.4153	−3.07205
$986$	0	0
$987$	−9.06070	−0.288405
$988$	0	0
$989$	−32.1659	−1.02282
$990$	0	0
$991$	38.7576	1.23117	0.615587	−	0.788069i	$-0.288919\pi$
$991$	38.7576	1.23117	0.615587	+	0.788069i	$0.288919\pi$
$992$	0	0
$993$	19.0400	0.604215
$994$	0	0
$995$	50.7467	1.60878
$996$	0	0
$997$	−6.43309	−0.203738	−0.101869	−	0.994798i	$-0.532482\pi$
$997$	−6.43309	−0.203738	−0.101869	+	0.994798i	$0.532482\pi$
$998$	0	0
$999$	3.37778	0.106868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.e.1.3	✓	3
4.3	odd	2		8016.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.e.1.3	✓	3	1.1	even	1	trivial
8016.2.a.n.1.3		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.21432 q^{5} -1.37778 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.21432 q^{5} -1.37778 q^{7} +1.00000 q^{9} -3.52543 q^{11} +2.00000 q^{13} -4.21432 q^{15} -6.64296 q^{17} +5.80642 q^{19} +1.37778 q^{21} -8.00000 q^{23} +12.7605 q^{25} -1.00000 q^{27} -9.05086 q^{29} -5.80642 q^{31} +3.52543 q^{33} -5.80642 q^{35} -3.37778 q^{37} -2.00000 q^{39} +10.7763 q^{41} +4.02074 q^{43} +4.21432 q^{45} -6.57628 q^{47} -5.10171 q^{49} +6.64296 q^{51} -9.39853 q^{53} -14.8573 q^{55} -5.80642 q^{57} -3.05086 q^{59} -10.5161 q^{61} -1.37778 q^{63} +8.42864 q^{65} -6.40790 q^{67} +8.00000 q^{69} +8.04149 q^{71} +16.2351 q^{73} -12.7605 q^{75} +4.85728 q^{77} -7.69381 q^{79} +1.00000 q^{81} -14.7971 q^{83} -27.9956 q^{85} +9.05086 q^{87} -6.29529 q^{89} -2.75557 q^{91} +5.80642 q^{93} +24.4701 q^{95} +15.2859 q^{97} -3.52543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 6 q^{15} + 4 q^{19} + 4 q^{21} - 24 q^{23} + 5 q^{25} - 3 q^{27} - 14 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{35} - 10 q^{37} - 6 q^{39} + 12 q^{41} - 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} - 18 q^{55} - 4 q^{57} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 12 q^{65} - 26 q^{67} + 24 q^{69} - 16 q^{71} + 22 q^{73} - 5 q^{75} - 12 q^{77} + 10 q^{79} + 3 q^{81} - 4 q^{83} - 24 q^{85} + 14 q^{87} - 6 q^{89} - 8 q^{91} + 4 q^{93} + 20 q^{95} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9 - 4 * q^11 + 6 * q^13 - 6 * q^15 + 4 * q^19 + 4 * q^21 - 24 * q^23 + 5 * q^25 - 3 * q^27 - 14 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 10 * q^37 - 6 * q^39 + 12 * q^41 - 8 * q^43 + 6 * q^45 + 11 * q^49 - 8 * q^53 - 18 * q^55 - 4 * q^57 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 12 * q^65 - 26 * q^67 + 24 * q^69 - 16 * q^71 + 22 * q^73 - 5 * q^75 - 12 * q^77 + 10 * q^79 + 3 * q^81 - 4 * q^83 - 24 * q^85 + 14 * q^87 - 6 * q^89 - 8 * q^91 + 4 * q^93 + 20 * q^95 + 6 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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