LMFDB - Embedded newform 4008.2.a.g.1.6

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 $ x^7 - 2x^6 - 7x^5 + 11x^4 + 12x^3 - 14x^2 - 6x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.20126$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.82640 q^{5} +2.26944 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.82640 q^{5} +2.26944 q^{7} +1.00000 q^{9} -0.576121 q^{11} -1.28343 q^{13} -1.82640 q^{15} -1.88405 q^{17} +5.66549 q^{19} -2.26944 q^{21} -2.15348 q^{23} -1.66425 q^{25} -1.00000 q^{27} +6.70504 q^{29} +2.07544 q^{31} +0.576121 q^{33} +4.14491 q^{35} -5.63675 q^{37} +1.28343 q^{39} +8.23751 q^{41} +2.54507 q^{43} +1.82640 q^{45} +9.37328 q^{47} -1.84966 q^{49} +1.88405 q^{51} +6.17753 q^{53} -1.05223 q^{55} -5.66549 q^{57} -3.25370 q^{59} +1.95266 q^{61} +2.26944 q^{63} -2.34406 q^{65} +8.37621 q^{67} +2.15348 q^{69} -2.61175 q^{71} -5.82577 q^{73} +1.66425 q^{75} -1.30747 q^{77} -1.10559 q^{79} +1.00000 q^{81} +14.4804 q^{83} -3.44103 q^{85} -6.70504 q^{87} -3.01816 q^{89} -2.91266 q^{91} -2.07544 q^{93} +10.3475 q^{95} -5.58912 q^{97} -0.576121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) $ 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9 + q^11 - 2 * q^13 + 3 * q^15 + 11 * q^17 + 2 * q^19 - 8 * q^21 + 17 * q^23 + 4 * q^25 - 7 * q^27 - 7 * q^29 + 10 * q^31 - q^33 + 10 * q^35 - 21 * q^37 + 2 * q^39 + 8 * q^41 - 12 * q^43 - 3 * q^45 + 25 * q^47 - 7 * q^49 - 11 * q^51 - 7 * q^53 + 15 * q^55 - 2 * q^57 + 3 * q^59 - 14 * q^61 + 8 * q^63 + 4 * q^65 + 4 * q^67 - 17 * q^69 + 27 * q^71 - 12 * q^73 - 4 * q^75 + 16 * q^77 + 8 * q^79 + 7 * q^81 + 15 * q^83 - 3 * q^85 + 7 * q^87 + 14 * q^89 - 3 * q^91 - 10 * q^93 + 37 * q^95 + 3 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.82640	0.816793	0.408396	−	0.912805i	$-0.366088\pi$
$5$	1.82640	0.816793	0.408396	+	0.912805i	$0.366088\pi$
$6$	0	0
$7$	2.26944	0.857766	0.428883	−	0.903360i	$-0.358907\pi$
$7$	2.26944	0.857766	0.428883	+	0.903360i	$0.358907\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.576121	−0.173707	−0.0868535	−	0.996221i	$-0.527681\pi$
$11$	−0.576121	−0.173707	−0.0868535	+	0.996221i	$0.527681\pi$
$12$	0	0
$13$	−1.28343	−0.355959	−0.177979	−	0.984034i	$-0.556956\pi$
$13$	−1.28343	−0.355959	−0.177979	+	0.984034i	$0.556956\pi$
$14$	0	0
$15$	−1.82640	−0.471576
$16$	0	0
$17$	−1.88405	−0.456949	−0.228474	−	0.973550i	$-0.573374\pi$
$17$	−1.88405	−0.456949	−0.228474	+	0.973550i	$0.573374\pi$
$18$	0	0
$19$	5.66549	1.29975	0.649876	−	0.760040i	$-0.274821\pi$
$19$	5.66549	1.29975	0.649876	+	0.760040i	$0.274821\pi$
$20$	0	0
$21$	−2.26944	−0.495231
$22$	0	0
$23$	−2.15348	−0.449032	−0.224516	−	0.974470i	$-0.572080\pi$
$23$	−2.15348	−0.449032	−0.224516	+	0.974470i	$0.572080\pi$
$24$	0	0
$25$	−1.66425	−0.332850
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.70504	1.24509	0.622547	−	0.782582i	$-0.286098\pi$
$29$	6.70504	1.24509	0.622547	+	0.782582i	$0.286098\pi$
$30$	0	0
$31$	2.07544	0.372761	0.186380	−	0.982478i	$-0.440324\pi$
$31$	2.07544	0.372761	0.186380	+	0.982478i	$0.440324\pi$
$32$	0	0
$33$	0.576121	0.100290
$34$	0	0
$35$	4.14491	0.700617
$36$	0	0
$37$	−5.63675	−0.926676	−0.463338	−	0.886182i	$-0.653348\pi$
$37$	−5.63675	−0.926676	−0.463338	+	0.886182i	$0.653348\pi$
$38$	0	0
$39$	1.28343	0.205513
$40$	0	0
$41$	8.23751	1.28648	0.643241	−	0.765664i	$-0.277589\pi$
$41$	8.23751	1.28648	0.643241	+	0.765664i	$0.277589\pi$
$42$	0	0
$43$	2.54507	0.388120	0.194060	−	0.980990i	$-0.437834\pi$
$43$	2.54507	0.388120	0.194060	+	0.980990i	$0.437834\pi$
$44$	0	0
$45$	1.82640	0.272264
$46$	0	0
$47$	9.37328	1.36723	0.683617	−	0.729841i	$-0.260406\pi$
$47$	9.37328	1.36723	0.683617	+	0.729841i	$0.260406\pi$
$48$	0	0
$49$	−1.84966	−0.264237
$50$	0	0
$51$	1.88405	0.263820
$52$	0	0
$53$	6.17753	0.848548	0.424274	−	0.905534i	$-0.360529\pi$
$53$	6.17753	0.848548	0.424274	+	0.905534i	$0.360529\pi$
$54$	0	0
$55$	−1.05223	−0.141883
$56$	0	0
$57$	−5.66549	−0.750413
$58$	0	0
$59$	−3.25370	−0.423596	−0.211798	−	0.977313i	$-0.567932\pi$
$59$	−3.25370	−0.423596	−0.211798	+	0.977313i	$0.567932\pi$
$60$	0	0
$61$	1.95266	0.250013	0.125006	−	0.992156i	$-0.460105\pi$
$61$	1.95266	0.250013	0.125006	+	0.992156i	$0.460105\pi$
$62$	0	0
$63$	2.26944	0.285922
$64$	0	0
$65$	−2.34406	−0.290745
$66$	0	0
$67$	8.37621	1.02332	0.511659	−	0.859189i	$-0.329031\pi$
$67$	8.37621	1.02332	0.511659	+	0.859189i	$0.329031\pi$
$68$	0	0
$69$	2.15348	0.259249
$70$	0	0
$71$	−2.61175	−0.309958	−0.154979	−	0.987918i	$-0.549531\pi$
$71$	−2.61175	−0.309958	−0.154979	+	0.987918i	$0.549531\pi$
$72$	0	0
$73$	−5.82577	−0.681855	−0.340928	−	0.940090i	$-0.610741\pi$
$73$	−5.82577	−0.681855	−0.340928	+	0.940090i	$0.610741\pi$
$74$	0	0
$75$	1.66425	0.192171
$76$	0	0
$77$	−1.30747	−0.149000
$78$	0	0
$79$	−1.10559	−0.124389	−0.0621944	−	0.998064i	$-0.519810\pi$
$79$	−1.10559	−0.124389	−0.0621944	+	0.998064i	$0.519810\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.4804	1.58943	0.794717	−	0.606980i	$-0.207619\pi$
$83$	14.4804	1.58943	0.794717	+	0.606980i	$0.207619\pi$
$84$	0	0
$85$	−3.44103	−0.373233
$86$	0	0
$87$	−6.70504	−0.718856
$88$	0	0
$89$	−3.01816	−0.319925	−0.159962	−	0.987123i	$-0.551137\pi$
$89$	−3.01816	−0.319925	−0.159962	+	0.987123i	$0.551137\pi$
$90$	0	0
$91$	−2.91266	−0.305329
$92$	0	0
$93$	−2.07544	−0.215214
$94$	0	0
$95$	10.3475	1.06163
$96$	0	0
$97$	−5.58912	−0.567490	−0.283745	−	0.958900i	$-0.591577\pi$
$97$	−5.58912	−0.567490	−0.283745	+	0.958900i	$0.591577\pi$
$98$	0	0
$99$	−0.576121	−0.0579023
$100$	0	0
$101$	−10.9923	−1.09377	−0.546887	−	0.837206i	$-0.684187\pi$
$101$	−10.9923	−1.09377	−0.546887	+	0.837206i	$0.684187\pi$
$102$	0	0
$103$	12.3818	1.22002	0.610009	−	0.792395i	$-0.291166\pi$
$103$	12.3818	1.22002	0.610009	+	0.792395i	$0.291166\pi$
$104$	0	0
$105$	−4.14491	−0.404501
$106$	0	0
$107$	−11.3065	−1.09304	−0.546519	−	0.837447i	$-0.684047\pi$
$107$	−11.3065	−1.09304	−0.546519	+	0.837447i	$0.684047\pi$
$108$	0	0
$109$	−4.32015	−0.413795	−0.206898	−	0.978363i	$-0.566337\pi$
$109$	−4.32015	−0.413795	−0.206898	+	0.978363i	$0.566337\pi$
$110$	0	0
$111$	5.63675	0.535017
$112$	0	0
$113$	11.4319	1.07542	0.537712	−	0.843128i	$-0.319289\pi$
$113$	11.4319	1.07542	0.537712	+	0.843128i	$0.319289\pi$
$114$	0	0
$115$	−3.93313	−0.366766
$116$	0	0
$117$	−1.28343	−0.118653
$118$	0	0
$119$	−4.27573	−0.391955
$120$	0	0
$121$	−10.6681	−0.969826
$122$	0	0
$123$	−8.23751	−0.742751
$124$	0	0
$125$	−12.1716	−1.08866
$126$	0	0
$127$	13.9407	1.23704	0.618519	−	0.785770i	$-0.287733\pi$
$127$	13.9407	1.23704	0.618519	+	0.785770i	$0.287733\pi$
$128$	0	0
$129$	−2.54507	−0.224081
$130$	0	0
$131$	10.1443	0.886308	0.443154	−	0.896445i	$-0.353859\pi$
$131$	10.1443	0.886308	0.443154	+	0.896445i	$0.353859\pi$
$132$	0	0
$133$	12.8575	1.11488
$134$	0	0
$135$	−1.82640	−0.157192
$136$	0	0
$137$	23.0801	1.97187	0.985933	−	0.167144i	$-0.0534544\pi$
$137$	23.0801	1.97187	0.985933	+	0.167144i	$0.0534544\pi$
$138$	0	0
$139$	−11.7976	−1.00066	−0.500331	−	0.865834i	$-0.666788\pi$
$139$	−11.7976	−1.00066	−0.500331	+	0.865834i	$0.666788\pi$
$140$	0	0
$141$	−9.37328	−0.789373
$142$	0	0
$143$	0.739410	0.0618325
$144$	0	0
$145$	12.2461	1.01698
$146$	0	0
$147$	1.84966	0.152557
$148$	0	0
$149$	−17.6449	−1.44553	−0.722764	−	0.691095i	$-0.757129\pi$
$149$	−17.6449	−1.44553	−0.722764	+	0.691095i	$0.757129\pi$
$150$	0	0
$151$	18.2039	1.48141	0.740707	−	0.671828i	$-0.234490\pi$
$151$	18.2039	1.48141	0.740707	+	0.671828i	$0.234490\pi$
$152$	0	0
$153$	−1.88405	−0.152316
$154$	0	0
$155$	3.79060	0.304468
$156$	0	0
$157$	15.6419	1.24836	0.624180	−	0.781280i	$-0.285433\pi$
$157$	15.6419	1.24836	0.624180	+	0.781280i	$0.285433\pi$
$158$	0	0
$159$	−6.17753	−0.489910
$160$	0	0
$161$	−4.88719	−0.385165
$162$	0	0
$163$	8.11923	0.635947	0.317974	−	0.948100i	$-0.396998\pi$
$163$	8.11923	0.635947	0.317974	+	0.948100i	$0.396998\pi$
$164$	0	0
$165$	1.05223	0.0819160
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.3528	−0.873293
$170$	0	0
$171$	5.66549	0.433251
$172$	0	0
$173$	−20.4282	−1.55312	−0.776562	−	0.630041i	$-0.783038\pi$
$173$	−20.4282	−1.55312	−0.776562	+	0.630041i	$0.783038\pi$
$174$	0	0
$175$	−3.77690	−0.285507
$176$	0	0
$177$	3.25370	0.244563
$178$	0	0
$179$	5.48020	0.409609	0.204805	−	0.978803i	$-0.434344\pi$
$179$	5.48020	0.409609	0.204805	+	0.978803i	$0.434344\pi$
$180$	0	0
$181$	−22.3021	−1.65770	−0.828851	−	0.559470i	$-0.811005\pi$
$181$	−22.3021	−1.65770	−0.828851	+	0.559470i	$0.811005\pi$
$182$	0	0
$183$	−1.95266	−0.144345
$184$	0	0
$185$	−10.2950	−0.756902
$186$	0	0
$187$	1.08544	0.0793752
$188$	0	0
$189$	−2.26944	−0.165077
$190$	0	0
$191$	2.13733	0.154652	0.0773261	−	0.997006i	$-0.475362\pi$
$191$	2.13733	0.154652	0.0773261	+	0.997006i	$0.475362\pi$
$192$	0	0
$193$	17.7899	1.28054	0.640272	−	0.768148i	$-0.278822\pi$
$193$	17.7899	1.28054	0.640272	+	0.768148i	$0.278822\pi$
$194$	0	0
$195$	2.34406	0.167861
$196$	0	0
$197$	23.9537	1.70663	0.853317	−	0.521393i	$-0.174587\pi$
$197$	23.9537	1.70663	0.853317	+	0.521393i	$0.174587\pi$
$198$	0	0
$199$	9.64250	0.683538	0.341769	−	0.939784i	$-0.388974\pi$
$199$	9.64250	0.683538	0.341769	+	0.939784i	$0.388974\pi$
$200$	0	0
$201$	−8.37621	−0.590813
$202$	0	0
$203$	15.2167	1.06800
$204$	0	0
$205$	15.0450	1.05079
$206$	0	0
$207$	−2.15348	−0.149677
$208$	0	0
$209$	−3.26401	−0.225776
$210$	0	0
$211$	0.232543	0.0160089	0.00800446	−	0.999968i	$-0.497452\pi$
$211$	0.232543	0.0160089	0.00800446	+	0.999968i	$0.497452\pi$
$212$	0	0
$213$	2.61175	0.178954
$214$	0	0
$215$	4.64833	0.317013
$216$	0	0
$217$	4.71009	0.319742
$218$	0	0
$219$	5.82577	0.393669
$220$	0	0
$221$	2.41804	0.162655
$222$	0	0
$223$	−21.2086	−1.42023	−0.710117	−	0.704084i	$-0.751358\pi$
$223$	−21.2086	−1.42023	−0.710117	+	0.704084i	$0.751358\pi$
$224$	0	0
$225$	−1.66425	−0.110950
$226$	0	0
$227$	−9.72014	−0.645148	−0.322574	−	0.946544i	$-0.604548\pi$
$227$	−9.72014	−0.645148	−0.322574	+	0.946544i	$0.604548\pi$
$228$	0	0
$229$	0.371580	0.0245547	0.0122773	−	0.999925i	$-0.496092\pi$
$229$	0.371580	0.0245547	0.0122773	+	0.999925i	$0.496092\pi$
$230$	0	0
$231$	1.30747	0.0860252
$232$	0	0
$233$	14.0679	0.921619	0.460810	−	0.887499i	$-0.347559\pi$
$233$	14.0679	0.921619	0.460810	+	0.887499i	$0.347559\pi$
$234$	0	0
$235$	17.1194	1.11675
$236$	0	0
$237$	1.10559	0.0718159
$238$	0	0
$239$	4.93902	0.319479	0.159739	−	0.987159i	$-0.448935\pi$
$239$	4.93902	0.319479	0.159739	+	0.987159i	$0.448935\pi$
$240$	0	0
$241$	−11.4721	−0.738985	−0.369493	−	0.929234i	$-0.620469\pi$
$241$	−11.4721	−0.738985	−0.369493	+	0.929234i	$0.620469\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.37823	−0.215827
$246$	0	0
$247$	−7.27125	−0.462658
$248$	0	0
$249$	−14.4804	−0.917661
$250$	0	0
$251$	20.6662	1.30444	0.652219	−	0.758031i	$-0.273838\pi$
$251$	20.6662	1.30444	0.652219	+	0.758031i	$0.273838\pi$
$252$	0	0
$253$	1.24067	0.0780001
$254$	0	0
$255$	3.44103	0.215486
$256$	0	0
$257$	24.4158	1.52302	0.761509	−	0.648155i	$-0.224459\pi$
$257$	24.4158	1.52302	0.761509	+	0.648155i	$0.224459\pi$
$258$	0	0
$259$	−12.7922	−0.794871
$260$	0	0
$261$	6.70504	0.415031
$262$	0	0
$263$	7.72951	0.476622	0.238311	−	0.971189i	$-0.423406\pi$
$263$	7.72951	0.476622	0.238311	+	0.971189i	$0.423406\pi$
$264$	0	0
$265$	11.2827	0.693088
$266$	0	0
$267$	3.01816	0.184709
$268$	0	0
$269$	11.1431	0.679409	0.339705	−	0.940532i	$-0.389673\pi$
$269$	11.1431	0.679409	0.339705	+	0.940532i	$0.389673\pi$
$270$	0	0
$271$	10.4906	0.637259	0.318630	−	0.947879i	$-0.396777\pi$
$271$	10.4906	0.637259	0.318630	+	0.947879i	$0.396777\pi$
$272$	0	0
$273$	2.91266	0.176282
$274$	0	0
$275$	0.958808	0.0578183
$276$	0	0
$277$	7.71364	0.463468	0.231734	−	0.972779i	$-0.425560\pi$
$277$	7.71364	0.463468	0.231734	+	0.972779i	$0.425560\pi$
$278$	0	0
$279$	2.07544	0.124254
$280$	0	0
$281$	26.5013	1.58093	0.790467	−	0.612504i	$-0.209838\pi$
$281$	26.5013	1.58093	0.790467	+	0.612504i	$0.209838\pi$
$282$	0	0
$283$	9.42532	0.560277	0.280139	−	0.959960i	$-0.409620\pi$
$283$	9.42532	0.560277	0.280139	+	0.959960i	$0.409620\pi$
$284$	0	0
$285$	−10.3475	−0.612932
$286$	0	0
$287$	18.6945	1.10350
$288$	0	0
$289$	−13.4504	−0.791198
$290$	0	0
$291$	5.58912	0.327640
$292$	0	0
$293$	−8.72067	−0.509467	−0.254734	−	0.967011i	$-0.581988\pi$
$293$	−8.72067	−0.509467	−0.254734	+	0.967011i	$0.581988\pi$
$294$	0	0
$295$	−5.94257	−0.345990
$296$	0	0
$297$	0.576121	0.0334299
$298$	0	0
$299$	2.76384	0.159837
$300$	0	0
$301$	5.77587	0.332916
$302$	0	0
$303$	10.9923	0.631491
$304$	0	0
$305$	3.56635	0.204209
$306$	0	0
$307$	−8.58502	−0.489973	−0.244986	−	0.969527i	$-0.578783\pi$
$307$	−8.58502	−0.489973	−0.244986	+	0.969527i	$0.578783\pi$
$308$	0	0
$309$	−12.3818	−0.704377
$310$	0	0
$311$	14.4072	0.816955	0.408478	−	0.912768i	$-0.366060\pi$
$311$	14.4072	0.816955	0.408478	+	0.912768i	$0.366060\pi$
$312$	0	0
$313$	−25.1446	−1.42126	−0.710628	−	0.703568i	$-0.751589\pi$
$313$	−25.1446	−1.42126	−0.710628	+	0.703568i	$0.751589\pi$
$314$	0	0
$315$	4.14491	0.233539
$316$	0	0
$317$	−5.08844	−0.285795	−0.142898	−	0.989737i	$-0.545642\pi$
$317$	−5.08844	−0.285795	−0.142898	+	0.989737i	$0.545642\pi$
$318$	0	0
$319$	−3.86291	−0.216282
$320$	0	0
$321$	11.3065	0.631065
$322$	0	0
$323$	−10.6741	−0.593920
$324$	0	0
$325$	2.13594	0.118481
$326$	0	0
$327$	4.32015	0.238905
$328$	0	0
$329$	21.2721	1.17277
$330$	0	0
$331$	21.6459	1.18976	0.594882	−	0.803813i	$-0.297199\pi$
$331$	21.6459	1.18976	0.594882	+	0.803813i	$0.297199\pi$
$332$	0	0
$333$	−5.63675	−0.308892
$334$	0	0
$335$	15.2984	0.835838
$336$	0	0
$337$	16.5621	0.902194	0.451097	−	0.892475i	$-0.351033\pi$
$337$	16.5621	0.902194	0.451097	+	0.892475i	$0.351033\pi$
$338$	0	0
$339$	−11.4319	−0.620897
$340$	0	0
$341$	−1.19571	−0.0647512
$342$	0	0
$343$	−20.0837	−1.08442
$344$	0	0
$345$	3.93313	0.211753
$346$	0	0
$347$	−9.57361	−0.513938	−0.256969	−	0.966420i	$-0.582724\pi$
$347$	−9.57361	−0.513938	−0.256969	+	0.966420i	$0.582724\pi$
$348$	0	0
$349$	−19.9918	−1.07014	−0.535068	−	0.844809i	$-0.679714\pi$
$349$	−19.9918	−1.07014	−0.535068	+	0.844809i	$0.679714\pi$
$350$	0	0
$351$	1.28343	0.0685043
$352$	0	0
$353$	21.3481	1.13624	0.568121	−	0.822945i	$-0.307671\pi$
$353$	21.3481	1.13624	0.568121	+	0.822945i	$0.307671\pi$
$354$	0	0
$355$	−4.77011	−0.253171
$356$	0	0
$357$	4.27573	0.226295
$358$	0	0
$359$	7.71901	0.407394	0.203697	−	0.979034i	$-0.434704\pi$
$359$	7.71901	0.407394	0.203697	+	0.979034i	$0.434704\pi$
$360$	0	0
$361$	13.0978	0.689357
$362$	0	0
$363$	10.6681	0.559929
$364$	0	0
$365$	−10.6402	−0.556934
$366$	0	0
$367$	−0.0577004	−0.00301194	−0.00150597	−	0.999999i	$-0.500479\pi$
$367$	−0.0577004	−0.00301194	−0.00150597	+	0.999999i	$0.500479\pi$
$368$	0	0
$369$	8.23751	0.428827
$370$	0	0
$371$	14.0195	0.727856
$372$	0	0
$373$	−9.25746	−0.479333	−0.239666	−	0.970855i	$-0.577038\pi$
$373$	−9.25746	−0.479333	−0.239666	+	0.970855i	$0.577038\pi$
$374$	0	0
$375$	12.1716	0.628539
$376$	0	0
$377$	−8.60543	−0.443202
$378$	0	0
$379$	5.75854	0.295796	0.147898	−	0.989003i	$-0.452749\pi$
$379$	5.75854	0.295796	0.147898	+	0.989003i	$0.452749\pi$
$380$	0	0
$381$	−13.9407	−0.714204
$382$	0	0
$383$	15.2551	0.779498	0.389749	−	0.920921i	$-0.372562\pi$
$383$	15.2551	0.779498	0.389749	+	0.920921i	$0.372562\pi$
$384$	0	0
$385$	−2.38797	−0.121702
$386$	0	0
$387$	2.54507	0.129373
$388$	0	0
$389$	3.29042	0.166831	0.0834156	−	0.996515i	$-0.473417\pi$
$389$	3.29042	0.166831	0.0834156	+	0.996515i	$0.473417\pi$
$390$	0	0
$391$	4.05727	0.205185
$392$	0	0
$393$	−10.1443	−0.511710
$394$	0	0
$395$	−2.01926	−0.101600
$396$	0	0
$397$	−4.37992	−0.219822	−0.109911	−	0.993941i	$-0.535057\pi$
$397$	−4.37992	−0.219822	−0.109911	+	0.993941i	$0.535057\pi$
$398$	0	0
$399$	−12.8575	−0.643678
$400$	0	0
$401$	−8.73266	−0.436088	−0.218044	−	0.975939i	$-0.569968\pi$
$401$	−8.73266	−0.436088	−0.218044	+	0.975939i	$0.569968\pi$
$402$	0	0
$403$	−2.66368	−0.132688
$404$	0	0
$405$	1.82640	0.0907548
$406$	0	0
$407$	3.24745	0.160970
$408$	0	0
$409$	36.6405	1.81176	0.905878	−	0.423538i	$-0.139212\pi$
$409$	36.6405	1.81176	0.905878	+	0.423538i	$0.139212\pi$
$410$	0	0
$411$	−23.0801	−1.13846
$412$	0	0
$413$	−7.38406	−0.363346
$414$	0	0
$415$	26.4471	1.29824
$416$	0	0
$417$	11.7976	0.577733
$418$	0	0
$419$	7.95948	0.388846	0.194423	−	0.980918i	$-0.437717\pi$
$419$	7.95948	0.388846	0.194423	+	0.980918i	$0.437717\pi$
$420$	0	0
$421$	−35.9794	−1.75353	−0.876765	−	0.480918i	$-0.840303\pi$
$421$	−35.9794	−1.75353	−0.876765	+	0.480918i	$0.840303\pi$
$422$	0	0
$423$	9.37328	0.455745
$424$	0	0
$425$	3.13552	0.152095
$426$	0	0
$427$	4.43144	0.214453
$428$	0	0
$429$	−0.739410	−0.0356990
$430$	0	0
$431$	26.4280	1.27299	0.636496	−	0.771280i	$-0.280383\pi$
$431$	26.4280	1.27299	0.636496	+	0.771280i	$0.280383\pi$
$432$	0	0
$433$	2.50877	0.120564	0.0602818	−	0.998181i	$-0.480800\pi$
$433$	2.50877	0.120564	0.0602818	+	0.998181i	$0.480800\pi$
$434$	0	0
$435$	−12.2461	−0.587156
$436$	0	0
$437$	−12.2005	−0.583631
$438$	0	0
$439$	−14.1782	−0.676691	−0.338345	−	0.941022i	$-0.609867\pi$
$439$	−14.1782	−0.676691	−0.338345	+	0.941022i	$0.609867\pi$
$440$	0	0
$441$	−1.84966	−0.0880791
$442$	0	0
$443$	5.51322	0.261941	0.130971	−	0.991386i	$-0.458191\pi$
$443$	5.51322	0.261941	0.130971	+	0.991386i	$0.458191\pi$
$444$	0	0
$445$	−5.51239	−0.261312
$446$	0	0
$447$	17.6449	0.834576
$448$	0	0
$449$	−33.5392	−1.58281	−0.791406	−	0.611290i	$-0.790651\pi$
$449$	−33.5392	−1.58281	−0.791406	+	0.611290i	$0.790651\pi$
$450$	0	0
$451$	−4.74580	−0.223471
$452$	0	0
$453$	−18.2039	−0.855295
$454$	0	0
$455$	−5.31969	−0.249391
$456$	0	0
$457$	19.1715	0.896806	0.448403	−	0.893831i	$-0.351993\pi$
$457$	19.1715	0.896806	0.448403	+	0.893831i	$0.351993\pi$
$458$	0	0
$459$	1.88405	0.0879399
$460$	0	0
$461$	20.4315	0.951588	0.475794	−	0.879557i	$-0.342161\pi$
$461$	20.4315	0.951588	0.475794	+	0.879557i	$0.342161\pi$
$462$	0	0
$463$	3.77055	0.175232	0.0876161	−	0.996154i	$-0.472075\pi$
$463$	3.77055	0.175232	0.0876161	+	0.996154i	$0.472075\pi$
$464$	0	0
$465$	−3.79060	−0.175785
$466$	0	0
$467$	3.99456	0.184846	0.0924230	−	0.995720i	$-0.470539\pi$
$467$	3.99456	0.184846	0.0924230	+	0.995720i	$0.470539\pi$
$468$	0	0
$469$	19.0093	0.877767
$470$	0	0
$471$	−15.6419	−0.720741
$472$	0	0
$473$	−1.46627	−0.0674191
$474$	0	0
$475$	−9.42878	−0.432622
$476$	0	0
$477$	6.17753	0.282849
$478$	0	0
$479$	−11.3905	−0.520445	−0.260222	−	0.965549i	$-0.583796\pi$
$479$	−11.3905	−0.520445	−0.260222	+	0.965549i	$0.583796\pi$
$480$	0	0
$481$	7.23436	0.329859
$482$	0	0
$483$	4.88719	0.222375
$484$	0	0
$485$	−10.2080	−0.463521
$486$	0	0
$487$	−30.4636	−1.38044	−0.690218	−	0.723601i	$-0.742485\pi$
$487$	−30.4636	−1.38044	−0.690218	+	0.723601i	$0.742485\pi$
$488$	0	0
$489$	−8.11923	−0.367164
$490$	0	0
$491$	−43.2815	−1.95327	−0.976633	−	0.214915i	$-0.931052\pi$
$491$	−43.2815	−1.95327	−0.976633	+	0.214915i	$0.931052\pi$
$492$	0	0
$493$	−12.6326	−0.568944
$494$	0	0
$495$	−1.05223	−0.0472942
$496$	0	0
$497$	−5.92720	−0.265871
$498$	0	0
$499$	−5.84147	−0.261500	−0.130750	−	0.991415i	$-0.541739\pi$
$499$	−5.84147	−0.261500	−0.130750	+	0.991415i	$0.541739\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−22.5856	−1.00704	−0.503520	−	0.863983i	$-0.667962\pi$
$503$	−22.5856	−1.00704	−0.503520	+	0.863983i	$0.667962\pi$
$504$	0	0
$505$	−20.0764	−0.893387
$506$	0	0
$507$	11.3528	0.504196
$508$	0	0
$509$	−5.82494	−0.258186	−0.129093	−	0.991633i	$-0.541207\pi$
$509$	−5.82494	−0.258186	−0.129093	+	0.991633i	$0.541207\pi$
$510$	0	0
$511$	−13.2212	−0.584872
$512$	0	0
$513$	−5.66549	−0.250138
$514$	0	0
$515$	22.6142	0.996501
$516$	0	0
$517$	−5.40015	−0.237498
$518$	0	0
$519$	20.4282	0.896697
$520$	0	0
$521$	2.08275	0.0912470	0.0456235	−	0.998959i	$-0.485473\pi$
$521$	2.08275	0.0912470	0.0456235	+	0.998959i	$0.485473\pi$
$522$	0	0
$523$	−1.94456	−0.0850296	−0.0425148	−	0.999096i	$-0.513537\pi$
$523$	−1.94456	−0.0850296	−0.0425148	+	0.999096i	$0.513537\pi$
$524$	0	0
$525$	3.77690	0.164838
$526$	0	0
$527$	−3.91024	−0.170333
$528$	0	0
$529$	−18.3625	−0.798370
$530$	0	0
$531$	−3.25370	−0.141199
$532$	0	0
$533$	−10.5722	−0.457935
$534$	0	0
$535$	−20.6502	−0.892785
$536$	0	0
$537$	−5.48020	−0.236488
$538$	0	0
$539$	1.06563	0.0458999
$540$	0	0
$541$	−26.7186	−1.14872	−0.574362	−	0.818601i	$-0.694750\pi$
$541$	−26.7186	−1.14872	−0.574362	+	0.818601i	$0.694750\pi$
$542$	0	0
$543$	22.3021	0.957074
$544$	0	0
$545$	−7.89034	−0.337985
$546$	0	0
$547$	−29.9878	−1.28218	−0.641092	−	0.767464i	$-0.721518\pi$
$547$	−29.9878	−1.28218	−0.641092	+	0.767464i	$0.721518\pi$
$548$	0	0
$549$	1.95266	0.0833376
$550$	0	0
$551$	37.9873	1.61831
$552$	0	0
$553$	−2.50907	−0.106697
$554$	0	0
$555$	10.2950	0.436998
$556$	0	0
$557$	−0.764104	−0.0323761	−0.0161881	−	0.999869i	$-0.505153\pi$
$557$	−0.764104	−0.0323761	−0.0161881	+	0.999869i	$0.505153\pi$
$558$	0	0
$559$	−3.26641	−0.138155
$560$	0	0
$561$	−1.08544	−0.0458273
$562$	0	0
$563$	−28.6526	−1.20756	−0.603781	−	0.797150i	$-0.706340\pi$
$563$	−28.6526	−1.20756	−0.603781	+	0.797150i	$0.706340\pi$
$564$	0	0
$565$	20.8793	0.878399
$566$	0	0
$567$	2.26944	0.0953073
$568$	0	0
$569$	39.4889	1.65546	0.827729	−	0.561127i	$-0.189632\pi$
$569$	39.4889	1.65546	0.827729	+	0.561127i	$0.189632\pi$
$570$	0	0
$571$	−39.5948	−1.65699	−0.828496	−	0.559995i	$-0.810803\pi$
$571$	−39.5948	−1.65699	−0.828496	+	0.559995i	$0.810803\pi$
$572$	0	0
$573$	−2.13733	−0.0892884
$574$	0	0
$575$	3.58393	0.149460
$576$	0	0
$577$	−16.3008	−0.678610	−0.339305	−	0.940676i	$-0.610192\pi$
$577$	−16.3008	−0.678610	−0.339305	+	0.940676i	$0.610192\pi$
$578$	0	0
$579$	−17.7899	−0.739322
$580$	0	0
$581$	32.8624	1.36336
$582$	0	0
$583$	−3.55900	−0.147399
$584$	0	0
$585$	−2.34406	−0.0969149
$586$	0	0
$587$	13.1281	0.541855	0.270927	−	0.962600i	$-0.412670\pi$
$587$	13.1281	0.541855	0.270927	+	0.962600i	$0.412670\pi$
$588$	0	0
$589$	11.7584	0.484497
$590$	0	0
$591$	−23.9537	−0.985325
$592$	0	0
$593$	4.17646	0.171507	0.0857533	−	0.996316i	$-0.472670\pi$
$593$	4.17646	0.171507	0.0857533	+	0.996316i	$0.472670\pi$
$594$	0	0
$595$	−7.80921	−0.320146
$596$	0	0
$597$	−9.64250	−0.394641
$598$	0	0
$599$	−39.9237	−1.63124	−0.815618	−	0.578590i	$-0.803603\pi$
$599$	−39.9237	−1.63124	−0.815618	+	0.578590i	$0.803603\pi$
$600$	0	0
$601$	−0.528562	−0.0215605	−0.0107803	−	0.999942i	$-0.503432\pi$
$601$	−0.528562	−0.0215605	−0.0107803	+	0.999942i	$0.503432\pi$
$602$	0	0
$603$	8.37621	0.341106
$604$	0	0
$605$	−19.4842	−0.792147
$606$	0	0
$607$	−11.2201	−0.455411	−0.227706	−	0.973730i	$-0.573122\pi$
$607$	−11.2201	−0.455411	−0.227706	+	0.973730i	$0.573122\pi$
$608$	0	0
$609$	−15.2167	−0.616610
$610$	0	0
$611$	−12.0299	−0.486679
$612$	0	0
$613$	24.9044	1.00588	0.502940	−	0.864321i	$-0.332252\pi$
$613$	24.9044	1.00588	0.502940	+	0.864321i	$0.332252\pi$
$614$	0	0
$615$	−15.0450	−0.606674
$616$	0	0
$617$	−6.39995	−0.257652	−0.128826	−	0.991667i	$-0.541121\pi$
$617$	−6.39995	−0.257652	−0.128826	+	0.991667i	$0.541121\pi$
$618$	0	0
$619$	−10.9438	−0.439869	−0.219935	−	0.975515i	$-0.570584\pi$
$619$	−10.9438	−0.439869	−0.219935	+	0.975515i	$0.570584\pi$
$620$	0	0
$621$	2.15348	0.0864163
$622$	0	0
$623$	−6.84953	−0.274421
$624$	0	0
$625$	−13.9090	−0.556362
$626$	0	0
$627$	3.26401	0.130352
$628$	0	0
$629$	10.6199	0.423444
$630$	0	0
$631$	15.6786	0.624155	0.312078	−	0.950057i	$-0.398975\pi$
$631$	15.6786	0.624155	0.312078	+	0.950057i	$0.398975\pi$
$632$	0	0
$633$	−0.232543	−0.00924275
$634$	0	0
$635$	25.4614	1.01040
$636$	0	0
$637$	2.37391	0.0940576
$638$	0	0
$639$	−2.61175	−0.103319
$640$	0	0
$641$	−0.388862	−0.0153591	−0.00767957	−	0.999971i	$-0.502445\pi$
$641$	−0.388862	−0.0153591	−0.00767957	+	0.999971i	$0.502445\pi$
$642$	0	0
$643$	22.2838	0.878787	0.439393	−	0.898295i	$-0.355193\pi$
$643$	22.2838	0.878787	0.439393	+	0.898295i	$0.355193\pi$
$644$	0	0
$645$	−4.64833	−0.183028
$646$	0	0
$647$	43.4363	1.70766	0.853829	−	0.520554i	$-0.174274\pi$
$647$	43.4363	1.70766	0.853829	+	0.520554i	$0.174274\pi$
$648$	0	0
$649$	1.87452	0.0735815
$650$	0	0
$651$	−4.71009	−0.184603
$652$	0	0
$653$	17.1896	0.672680	0.336340	−	0.941741i	$-0.390811\pi$
$653$	17.1896	0.672680	0.336340	+	0.941741i	$0.390811\pi$
$654$	0	0
$655$	18.5275	0.723930
$656$	0	0
$657$	−5.82577	−0.227285
$658$	0	0
$659$	−4.43602	−0.172803	−0.0864015	−	0.996260i	$-0.527537\pi$
$659$	−4.43602	−0.172803	−0.0864015	+	0.996260i	$0.527537\pi$
$660$	0	0
$661$	46.9317	1.82543	0.912715	−	0.408597i	$-0.133982\pi$
$661$	46.9317	1.82543	0.912715	+	0.408597i	$0.133982\pi$
$662$	0	0
$663$	−2.41804	−0.0939089
$664$	0	0
$665$	23.4829	0.910629
$666$	0	0
$667$	−14.4392	−0.559088
$668$	0	0
$669$	21.2086	0.819973
$670$	0	0
$671$	−1.12497	−0.0434290
$672$	0	0
$673$	−41.9815	−1.61827	−0.809134	−	0.587624i	$-0.800063\pi$
$673$	−41.9815	−1.61827	−0.809134	+	0.587624i	$0.800063\pi$
$674$	0	0
$675$	1.66425	0.0640569
$676$	0	0
$677$	−15.9875	−0.614449	−0.307224	−	0.951637i	$-0.599400\pi$
$677$	−15.9875	−0.614449	−0.307224	+	0.951637i	$0.599400\pi$
$678$	0	0
$679$	−12.6842	−0.486773
$680$	0	0
$681$	9.72014	0.372476
$682$	0	0
$683$	−1.80502	−0.0690673	−0.0345337	−	0.999404i	$-0.510995\pi$
$683$	−1.80502	−0.0690673	−0.0345337	+	0.999404i	$0.510995\pi$
$684$	0	0
$685$	42.1536	1.61061
$686$	0	0
$687$	−0.371580	−0.0141767
$688$	0	0
$689$	−7.92841	−0.302048
$690$	0	0
$691$	16.6566	0.633645	0.316823	−	0.948485i	$-0.397384\pi$
$691$	16.6566	0.633645	0.316823	+	0.948485i	$0.397384\pi$
$692$	0	0
$693$	−1.30747	−0.0496667
$694$	0	0
$695$	−21.5472	−0.817334
$696$	0	0
$697$	−15.5199	−0.587857
$698$	0	0
$699$	−14.0679	−0.532097
$700$	0	0
$701$	−42.6642	−1.61140	−0.805702	−	0.592321i	$-0.798212\pi$
$701$	−42.6642	−1.61140	−0.805702	+	0.592321i	$0.798212\pi$
$702$	0	0
$703$	−31.9350	−1.20445
$704$	0	0
$705$	−17.1194	−0.644754
$706$	0	0
$707$	−24.9463	−0.938203
$708$	0	0
$709$	4.94574	0.185741	0.0928706	−	0.995678i	$-0.470396\pi$
$709$	4.94574	0.185741	0.0928706	+	0.995678i	$0.470396\pi$
$710$	0	0
$711$	−1.10559	−0.0414630
$712$	0	0
$713$	−4.46944	−0.167382
$714$	0	0
$715$	1.35046	0.0505044
$716$	0	0
$717$	−4.93902	−0.184451
$718$	0	0
$719$	14.4179	0.537696	0.268848	−	0.963183i	$-0.413357\pi$
$719$	14.4179	0.537696	0.268848	+	0.963183i	$0.413357\pi$
$720$	0	0
$721$	28.0998	1.04649
$722$	0	0
$723$	11.4721	0.426653
$724$	0	0
$725$	−11.1588	−0.414429
$726$	0	0
$727$	7.72774	0.286606	0.143303	−	0.989679i	$-0.454228\pi$
$727$	7.72774	0.286606	0.143303	+	0.989679i	$0.454228\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.79504	−0.177351
$732$	0	0
$733$	−2.86167	−0.105698	−0.0528491	−	0.998603i	$-0.516830\pi$
$733$	−2.86167	−0.105698	−0.0528491	+	0.998603i	$0.516830\pi$
$734$	0	0
$735$	3.37823	0.124608
$736$	0	0
$737$	−4.82571	−0.177757
$738$	0	0
$739$	−44.8796	−1.65092	−0.825461	−	0.564459i	$-0.809085\pi$
$739$	−44.8796	−1.65092	−0.825461	+	0.564459i	$0.809085\pi$
$740$	0	0
$741$	7.27125	0.267116
$742$	0	0
$743$	−32.0693	−1.17651	−0.588255	−	0.808675i	$-0.700185\pi$
$743$	−32.0693	−1.17651	−0.588255	+	0.808675i	$0.700185\pi$
$744$	0	0
$745$	−32.2268	−1.18070
$746$	0	0
$747$	14.4804	0.529812
$748$	0	0
$749$	−25.6593	−0.937570
$750$	0	0
$751$	−30.4610	−1.11154	−0.555769	−	0.831337i	$-0.687576\pi$
$751$	−30.4610	−1.11154	−0.555769	+	0.831337i	$0.687576\pi$
$752$	0	0
$753$	−20.6662	−0.753117
$754$	0	0
$755$	33.2477	1.21001
$756$	0	0
$757$	−9.07724	−0.329918	−0.164959	−	0.986300i	$-0.552749\pi$
$757$	−9.07724	−0.329918	−0.164959	+	0.986300i	$0.552749\pi$
$758$	0	0
$759$	−1.24067	−0.0450334
$760$	0	0
$761$	−45.4211	−1.64651	−0.823256	−	0.567670i	$-0.807845\pi$
$761$	−45.4211	−1.64651	−0.823256	+	0.567670i	$0.807845\pi$
$762$	0	0
$763$	−9.80431	−0.354940
$764$	0	0
$765$	−3.44103	−0.124411
$766$	0	0
$767$	4.17589	0.150783
$768$	0	0
$769$	−37.9043	−1.36686	−0.683432	−	0.730014i	$-0.739514\pi$
$769$	−37.9043	−1.36686	−0.683432	+	0.730014i	$0.739514\pi$
$770$	0	0
$771$	−24.4158	−0.879315
$772$	0	0
$773$	−10.3435	−0.372030	−0.186015	−	0.982547i	$-0.559557\pi$
$773$	−10.3435	−0.372030	−0.186015	+	0.982547i	$0.559557\pi$
$774$	0	0
$775$	−3.45406	−0.124073
$776$	0	0
$777$	12.7922	0.458919
$778$	0	0
$779$	46.6695	1.67211
$780$	0	0
$781$	1.50468	0.0538418
$782$	0	0
$783$	−6.70504	−0.239619
$784$	0	0
$785$	28.5684	1.01965
$786$	0	0
$787$	−30.9881	−1.10461	−0.552304	−	0.833643i	$-0.686251\pi$
$787$	−30.9881	−1.10461	−0.552304	+	0.833643i	$0.686251\pi$
$788$	0	0
$789$	−7.72951	−0.275178
$790$	0	0
$791$	25.9440	0.922462
$792$	0	0
$793$	−2.50610	−0.0889943
$794$	0	0
$795$	−11.2827	−0.400155
$796$	0	0
$797$	−43.3862	−1.53682	−0.768409	−	0.639960i	$-0.778951\pi$
$797$	−43.3862	−1.53682	−0.768409	+	0.639960i	$0.778951\pi$
$798$	0	0
$799$	−17.6597	−0.624756
$800$	0	0
$801$	−3.01816	−0.106642
$802$	0	0
$803$	3.35635	0.118443
$804$	0	0
$805$	−8.92599	−0.314600
$806$	0	0
$807$	−11.1431	−0.392257
$808$	0	0
$809$	31.5421	1.10896	0.554480	−	0.832197i	$-0.312917\pi$
$809$	31.5421	1.10896	0.554480	+	0.832197i	$0.312917\pi$
$810$	0	0
$811$	−0.0655369	−0.00230131	−0.00115066	−	0.999999i	$-0.500366\pi$
$811$	−0.0655369	−0.00230131	−0.00115066	+	0.999999i	$0.500366\pi$
$812$	0	0
$813$	−10.4906	−0.367922
$814$	0	0
$815$	14.8290	0.519437
$816$	0	0
$817$	14.4191	0.504459
$818$	0	0
$819$	−2.91266	−0.101776
$820$	0	0
$821$	−6.79246	−0.237058	−0.118529	−	0.992951i	$-0.537818\pi$
$821$	−6.79246	−0.237058	−0.118529	+	0.992951i	$0.537818\pi$
$822$	0	0
$823$	−4.87356	−0.169882	−0.0849408	−	0.996386i	$-0.527070\pi$
$823$	−4.87356	−0.169882	−0.0849408	+	0.996386i	$0.527070\pi$
$824$	0	0
$825$	−0.958808	−0.0333814
$826$	0	0
$827$	31.0388	1.07933	0.539663	−	0.841881i	$-0.318552\pi$
$827$	31.0388	1.07933	0.539663	+	0.841881i	$0.318552\pi$
$828$	0	0
$829$	23.2306	0.806834	0.403417	−	0.915016i	$-0.367822\pi$
$829$	23.2306	0.806834	0.403417	+	0.915016i	$0.367822\pi$
$830$	0	0
$831$	−7.71364	−0.267583
$832$	0	0
$833$	3.48485	0.120743
$834$	0	0
$835$	−1.82640	−0.0632053
$836$	0	0
$837$	−2.07544	−0.0717379
$838$	0	0
$839$	−2.35242	−0.0812146	−0.0406073	−	0.999175i	$-0.512929\pi$
$839$	−2.35242	−0.0812146	−0.0406073	+	0.999175i	$0.512929\pi$
$840$	0	0
$841$	15.9575	0.550260
$842$	0	0
$843$	−26.5013	−0.912753
$844$	0	0
$845$	−20.7348	−0.713300
$846$	0	0
$847$	−24.2105	−0.831884
$848$	0	0
$849$	−9.42532	−0.323476
$850$	0	0
$851$	12.1387	0.416108
$852$	0	0
$853$	−51.6691	−1.76912	−0.884558	−	0.466430i	$-0.845540\pi$
$853$	−51.6691	−1.76912	−0.884558	+	0.466430i	$0.845540\pi$
$854$	0	0
$855$	10.3475	0.353876
$856$	0	0
$857$	−6.17501	−0.210934	−0.105467	−	0.994423i	$-0.533634\pi$
$857$	−6.17501	−0.210934	−0.105467	+	0.994423i	$0.533634\pi$
$858$	0	0
$859$	−28.9287	−0.987034	−0.493517	−	0.869736i	$-0.664289\pi$
$859$	−28.9287	−0.987034	−0.493517	+	0.869736i	$0.664289\pi$
$860$	0	0
$861$	−18.6945	−0.637107
$862$	0	0
$863$	9.28425	0.316039	0.158020	−	0.987436i	$-0.449489\pi$
$863$	9.28425	0.316039	0.158020	+	0.987436i	$0.449489\pi$
$864$	0	0
$865$	−37.3101	−1.26858
$866$	0	0
$867$	13.4504	0.456798
$868$	0	0
$869$	0.636955	0.0216072
$870$	0	0
$871$	−10.7503	−0.364259
$872$	0	0
$873$	−5.58912	−0.189163
$874$	0	0
$875$	−27.6227	−0.933817
$876$	0	0
$877$	14.8580	0.501718	0.250859	−	0.968024i	$-0.419287\pi$
$877$	14.8580	0.501718	0.250859	+	0.968024i	$0.419287\pi$
$878$	0	0
$879$	8.72067	0.294141
$880$	0	0
$881$	−23.8922	−0.804947	−0.402474	−	0.915432i	$-0.631850\pi$
$881$	−23.8922	−0.804947	−0.402474	+	0.915432i	$0.631850\pi$
$882$	0	0
$883$	−56.7643	−1.91027	−0.955135	−	0.296170i	$-0.904291\pi$
$883$	−56.7643	−1.91027	−0.955135	+	0.296170i	$0.904291\pi$
$884$	0	0
$885$	5.94257	0.199757
$886$	0	0
$887$	42.4315	1.42471	0.712355	−	0.701820i	$-0.247629\pi$
$887$	42.4315	1.42471	0.712355	+	0.701820i	$0.247629\pi$
$888$	0	0
$889$	31.6375	1.06109
$890$	0	0
$891$	−0.576121	−0.0193008
$892$	0	0
$893$	53.1043	1.77707
$894$	0	0
$895$	10.0091	0.334566
$896$	0	0
$897$	−2.76384	−0.0922820
$898$	0	0
$899$	13.9159	0.464122
$900$	0	0
$901$	−11.6388	−0.387743
$902$	0	0
$903$	−5.77587	−0.192209
$904$	0	0
$905$	−40.7326	−1.35400
$906$	0	0
$907$	42.7520	1.41956	0.709779	−	0.704424i	$-0.248795\pi$
$907$	42.7520	1.41956	0.709779	+	0.704424i	$0.248795\pi$
$908$	0	0
$909$	−10.9923	−0.364592
$910$	0	0
$911$	−4.12245	−0.136583	−0.0682914	−	0.997665i	$-0.521755\pi$
$911$	−4.12245	−0.136583	−0.0682914	+	0.997665i	$0.521755\pi$
$912$	0	0
$913$	−8.34248	−0.276096
$914$	0	0
$915$	−3.56635	−0.117900
$916$	0	0
$917$	23.0217	0.760245
$918$	0	0
$919$	−15.5909	−0.514295	−0.257148	−	0.966372i	$-0.582783\pi$
$919$	−15.5909	−0.514295	−0.257148	+	0.966372i	$0.582783\pi$
$920$	0	0
$921$	8.58502	0.282886
$922$	0	0
$923$	3.35199	0.110332
$924$	0	0
$925$	9.38095	0.308444
$926$	0	0
$927$	12.3818	0.406672
$928$	0	0
$929$	−35.6152	−1.16850	−0.584248	−	0.811575i	$-0.698611\pi$
$929$	−35.6152	−1.16850	−0.584248	+	0.811575i	$0.698611\pi$
$930$	0	0
$931$	−10.4792	−0.343443
$932$	0	0
$933$	−14.4072	−0.471669
$934$	0	0
$935$	1.98245	0.0648331
$936$	0	0
$937$	−5.39850	−0.176361	−0.0881806	−	0.996104i	$-0.528105\pi$
$937$	−5.39850	−0.176361	−0.0881806	+	0.996104i	$0.528105\pi$
$938$	0	0
$939$	25.1446	0.820563
$940$	0	0
$941$	−19.2566	−0.627747	−0.313873	−	0.949465i	$-0.601627\pi$
$941$	−19.2566	−0.627747	−0.313873	+	0.949465i	$0.601627\pi$
$942$	0	0
$943$	−17.7393	−0.577672
$944$	0	0
$945$	−4.14491	−0.134834
$946$	0	0
$947$	−12.0455	−0.391425	−0.195712	−	0.980661i	$-0.562702\pi$
$947$	−12.0455	−0.391425	−0.195712	+	0.980661i	$0.562702\pi$
$948$	0	0
$949$	7.47696	0.242712
$950$	0	0
$951$	5.08844	0.165004
$952$	0	0
$953$	−24.9635	−0.808647	−0.404324	−	0.914616i	$-0.632493\pi$
$953$	−24.9635	−0.808647	−0.404324	+	0.914616i	$0.632493\pi$
$954$	0	0
$955$	3.90364	0.126319
$956$	0	0
$957$	3.86291	0.124870
$958$	0	0
$959$	52.3788	1.69140
$960$	0	0
$961$	−26.6925	−0.861049
$962$	0	0
$963$	−11.3065	−0.364346
$964$	0	0
$965$	32.4915	1.04594
$966$	0	0
$967$	18.5543	0.596667	0.298333	−	0.954462i	$-0.403569\pi$
$967$	18.5543	0.596667	0.298333	+	0.954462i	$0.403569\pi$
$968$	0	0
$969$	10.6741	0.342900
$970$	0	0
$971$	−12.8450	−0.412214	−0.206107	−	0.978529i	$-0.566080\pi$
$971$	−12.8450	−0.412214	−0.206107	+	0.978529i	$0.566080\pi$
$972$	0	0
$973$	−26.7740	−0.858334
$974$	0	0
$975$	−2.13594	−0.0684049
$976$	0	0
$977$	46.2296	1.47902	0.739508	−	0.673148i	$-0.235058\pi$
$977$	46.2296	1.47902	0.739508	+	0.673148i	$0.235058\pi$
$978$	0	0
$979$	1.73883	0.0555732
$980$	0	0
$981$	−4.32015	−0.137932
$982$	0	0
$983$	−8.20446	−0.261682	−0.130841	−	0.991403i	$-0.541768\pi$
$983$	−8.20446	−0.261682	−0.130841	+	0.991403i	$0.541768\pi$
$984$	0	0
$985$	43.7492	1.39397
$986$	0	0
$987$	−21.2721	−0.677097
$988$	0	0
$989$	−5.48077	−0.174278
$990$	0	0
$991$	19.6139	0.623056	0.311528	−	0.950237i	$-0.399159\pi$
$991$	19.6139	0.623056	0.311528	+	0.950237i	$0.399159\pi$
$992$	0	0
$993$	−21.6459	−0.686911
$994$	0	0
$995$	17.6111	0.558309
$996$	0	0
$997$	3.59543	0.113868	0.0569342	−	0.998378i	$-0.481867\pi$
$997$	3.59543	0.113868	0.0569342	+	0.998378i	$0.481867\pi$
$998$	0	0
$999$	5.63675	0.178339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.g.1.6	✓	7
4.3	odd	2		8016.2.a.w.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.g.1.6	✓	7	1.1	even	1	trivial
8016.2.a.w.1.6		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.82640 q^{5} +2.26944 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.82640 q^{5} +2.26944 q^{7} +1.00000 q^{9} -0.576121 q^{11} -1.28343 q^{13} -1.82640 q^{15} -1.88405 q^{17} +5.66549 q^{19} -2.26944 q^{21} -2.15348 q^{23} -1.66425 q^{25} -1.00000 q^{27} +6.70504 q^{29} +2.07544 q^{31} +0.576121 q^{33} +4.14491 q^{35} -5.63675 q^{37} +1.28343 q^{39} +8.23751 q^{41} +2.54507 q^{43} +1.82640 q^{45} +9.37328 q^{47} -1.84966 q^{49} +1.88405 q^{51} +6.17753 q^{53} -1.05223 q^{55} -5.66549 q^{57} -3.25370 q^{59} +1.95266 q^{61} +2.26944 q^{63} -2.34406 q^{65} +8.37621 q^{67} +2.15348 q^{69} -2.61175 q^{71} -5.82577 q^{73} +1.66425 q^{75} -1.30747 q^{77} -1.10559 q^{79} +1.00000 q^{81} +14.4804 q^{83} -3.44103 q^{85} -6.70504 q^{87} -3.01816 q^{89} -2.91266 q^{91} -2.07544 q^{93} +10.3475 q^{95} -5.58912 q^{97} -0.576121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) \) 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9 + q^11 - 2 * q^13 + 3 * q^15 + 11 * q^17 + 2 * q^19 - 8 * q^21 + 17 * q^23 + 4 * q^25 - 7 * q^27 - 7 * q^29 + 10 * q^31 - q^33 + 10 * q^35 - 21 * q^37 + 2 * q^39 + 8 * q^41 - 12 * q^43 - 3 * q^45 + 25 * q^47 - 7 * q^49 - 11 * q^51 - 7 * q^53 + 15 * q^55 - 2 * q^57 + 3 * q^59 - 14 * q^61 + 8 * q^63 + 4 * q^65 + 4 * q^67 - 17 * q^69 + 27 * q^71 - 12 * q^73 - 4 * q^75 + 16 * q^77 + 8 * q^79 + 7 * q^81 + 15 * q^83 - 3 * q^85 + 7 * q^87 + 14 * q^89 - 3 * q^91 - 10 * q^93 + 37 * q^95 + 3 * q^97 + q^99

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