LMFDB - Embedded newform 4008.2.a.k.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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.0427374$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.957263 q^{5} -0.898491 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.957263 q^{5} -0.898491 q^{7} +1.00000 q^{9} -0.936680 q^{11} +1.86614 q^{13} -0.957263 q^{15} +6.88802 q^{17} +0.644628 q^{19} +0.898491 q^{21} -1.95775 q^{23} -4.08365 q^{25} -1.00000 q^{27} -0.602379 q^{29} +3.41687 q^{31} +0.936680 q^{33} -0.860092 q^{35} +11.6974 q^{37} -1.86614 q^{39} +0.378801 q^{41} -8.52617 q^{43} +0.957263 q^{45} -1.26703 q^{47} -6.19271 q^{49} -6.88802 q^{51} -9.54479 q^{53} -0.896648 q^{55} -0.644628 q^{57} -1.15472 q^{59} +12.4961 q^{61} -0.898491 q^{63} +1.78639 q^{65} +7.35619 q^{67} +1.95775 q^{69} +12.2991 q^{71} +0.169007 q^{73} +4.08365 q^{75} +0.841599 q^{77} -3.88101 q^{79} +1.00000 q^{81} -3.18308 q^{83} +6.59364 q^{85} +0.602379 q^{87} +6.13364 q^{89} -1.67671 q^{91} -3.41687 q^{93} +0.617078 q^{95} +6.69554 q^{97} -0.936680 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9	$ 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9} - q^{11} + 10 q^{13} - 10 q^{15} + 17 q^{17} + 2 q^{19} + q^{21} - 3 q^{23} + 21 q^{25} - 11 q^{27} + 17 q^{29} - 15 q^{31} + q^{33} + 11 q^{35} + 4 q^{37} - 10 q^{39} + 16 q^{41} + 10 q^{43} + 10 q^{45} - 16 q^{47} + 22 q^{49} - 17 q^{51} + 42 q^{53} - 5 q^{55} - 2 q^{57} - 2 q^{59} + 12 q^{61} - q^{63} + 10 q^{65} - q^{67} + 3 q^{69} - 9 q^{71} + 24 q^{73} - 21 q^{75} + 22 q^{77} - 30 q^{79} + 11 q^{81} + 16 q^{83} + 25 q^{85} - 17 q^{87} + 37 q^{89} + q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) $ 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9 - q^11 + 10 * q^13 - 10 * q^15 + 17 * q^17 + 2 * q^19 + q^21 - 3 * q^23 + 21 * q^25 - 11 * q^27 + 17 * q^29 - 15 * q^31 + q^33 + 11 * q^35 + 4 * q^37 - 10 * q^39 + 16 * q^41 + 10 * q^43 + 10 * q^45 - 16 * q^47 + 22 * q^49 - 17 * q^51 + 42 * q^53 - 5 * q^55 - 2 * q^57 - 2 * q^59 + 12 * q^61 - q^63 + 10 * q^65 - q^67 + 3 * q^69 - 9 * q^71 + 24 * q^73 - 21 * q^75 + 22 * q^77 - 30 * q^79 + 11 * q^81 + 16 * q^83 + 25 * q^85 - 17 * q^87 + 37 * q^89 + q^91 + 15 * q^93 + 5 * q^95 + 4 * 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
$5$	0.957263	0.428101	0.214050	−	0.976823i	$-0.431334\pi$
$5$	0.957263	0.428101	0.214050	+	0.976823i	$0.431334\pi$
$6$	0	0
$7$	−0.898491	−0.339598	−0.169799	−	0.985479i	$-0.554312\pi$
$7$	−0.898491	−0.339598	−0.169799	+	0.985479i	$0.554312\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.936680	−0.282420	−0.141210	−	0.989980i	$-0.545099\pi$
$11$	−0.936680	−0.282420	−0.141210	+	0.989980i	$0.545099\pi$
$12$	0	0
$13$	1.86614	0.517575	0.258788	−	0.965934i	$-0.416677\pi$
$13$	1.86614	0.517575	0.258788	+	0.965934i	$0.416677\pi$
$14$	0	0
$15$	−0.957263	−0.247164
$16$	0	0
$17$	6.88802	1.67059	0.835295	−	0.549802i	$-0.185297\pi$
$17$	6.88802	1.67059	0.835295	+	0.549802i	$0.185297\pi$
$18$	0	0
$19$	0.644628	0.147888	0.0739439	−	0.997262i	$-0.476441\pi$
$19$	0.644628	0.147888	0.0739439	+	0.997262i	$0.476441\pi$
$20$	0	0
$21$	0.898491	0.196067
$22$	0	0
$23$	−1.95775	−0.408219	−0.204110	−	0.978948i	$-0.565430\pi$
$23$	−1.95775	−0.408219	−0.204110	+	0.978948i	$0.565430\pi$
$24$	0	0
$25$	−4.08365	−0.816730
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.602379	−0.111859	−0.0559295	−	0.998435i	$-0.517812\pi$
$29$	−0.602379	−0.111859	−0.0559295	+	0.998435i	$0.517812\pi$
$30$	0	0
$31$	3.41687	0.613687	0.306844	−	0.951760i	$-0.400727\pi$
$31$	3.41687	0.613687	0.306844	+	0.951760i	$0.400727\pi$
$32$	0	0
$33$	0.936680	0.163055
$34$	0	0
$35$	−0.860092	−0.145382
$36$	0	0
$37$	11.6974	1.92304	0.961521	−	0.274731i	$-0.0885890\pi$
$37$	11.6974	1.92304	0.961521	+	0.274731i	$0.0885890\pi$
$38$	0	0
$39$	−1.86614	−0.298822
$40$	0	0
$41$	0.378801	0.0591588	0.0295794	−	0.999562i	$-0.490583\pi$
$41$	0.378801	0.0591588	0.0295794	+	0.999562i	$0.490583\pi$
$42$	0	0
$43$	−8.52617	−1.30023	−0.650114	−	0.759836i	$-0.725279\pi$
$43$	−8.52617	−1.30023	−0.650114	+	0.759836i	$0.725279\pi$
$44$	0	0
$45$	0.957263	0.142700
$46$	0	0
$47$	−1.26703	−0.184815	−0.0924075	−	0.995721i	$-0.529456\pi$
$47$	−1.26703	−0.184815	−0.0924075	+	0.995721i	$0.529456\pi$
$48$	0	0
$49$	−6.19271	−0.884673
$50$	0	0
$51$	−6.88802	−0.964516
$52$	0	0
$53$	−9.54479	−1.31108	−0.655539	−	0.755161i	$-0.727559\pi$
$53$	−9.54479	−1.31108	−0.655539	+	0.755161i	$0.727559\pi$
$54$	0	0
$55$	−0.896648	−0.120904
$56$	0	0
$57$	−0.644628	−0.0853830
$58$	0	0
$59$	−1.15472	−0.150331	−0.0751657	−	0.997171i	$-0.523949\pi$
$59$	−1.15472	−0.150331	−0.0751657	+	0.997171i	$0.523949\pi$
$60$	0	0
$61$	12.4961	1.59997	0.799984	−	0.600021i	$-0.204841\pi$
$61$	12.4961	1.59997	0.799984	+	0.600021i	$0.204841\pi$
$62$	0	0
$63$	−0.898491	−0.113199
$64$	0	0
$65$	1.78639	0.221574
$66$	0	0
$67$	7.35619	0.898702	0.449351	−	0.893355i	$-0.351655\pi$
$67$	7.35619	0.898702	0.449351	+	0.893355i	$0.351655\pi$
$68$	0	0
$69$	1.95775	0.235686
$70$	0	0
$71$	12.2991	1.45964	0.729819	−	0.683640i	$-0.239604\pi$
$71$	12.2991	1.45964	0.729819	+	0.683640i	$0.239604\pi$
$72$	0	0
$73$	0.169007	0.0197808	0.00989041	−	0.999951i	$-0.496852\pi$
$73$	0.169007	0.0197808	0.00989041	+	0.999951i	$0.496852\pi$
$74$	0	0
$75$	4.08365	0.471539
$76$	0	0
$77$	0.841599	0.0959091
$78$	0	0
$79$	−3.88101	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$79$	−3.88101	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.18308	−0.349388	−0.174694	−	0.984623i	$-0.555894\pi$
$83$	−3.18308	−0.349388	−0.174694	+	0.984623i	$0.555894\pi$
$84$	0	0
$85$	6.59364	0.715181
$86$	0	0
$87$	0.602379	0.0645818
$88$	0	0
$89$	6.13364	0.650165	0.325082	−	0.945686i	$-0.394608\pi$
$89$	6.13364	0.650165	0.325082	+	0.945686i	$0.394608\pi$
$90$	0	0
$91$	−1.67671	−0.175767
$92$	0	0
$93$	−3.41687	−0.354313
$94$	0	0
$95$	0.617078	0.0633109
$96$	0	0
$97$	6.69554	0.679829	0.339914	−	0.940456i	$-0.389602\pi$
$97$	6.69554	0.679829	0.339914	+	0.940456i	$0.389602\pi$
$98$	0	0
$99$	−0.936680	−0.0941398
$100$	0	0
$101$	−1.63976	−0.163162	−0.0815812	−	0.996667i	$-0.525997\pi$
$101$	−1.63976	−0.163162	−0.0815812	+	0.996667i	$0.525997\pi$
$102$	0	0
$103$	−15.3568	−1.51315	−0.756575	−	0.653907i	$-0.773129\pi$
$103$	−15.3568	−1.51315	−0.756575	+	0.653907i	$0.773129\pi$
$104$	0	0
$105$	0.860092	0.0839364
$106$	0	0
$107$	17.5800	1.69952	0.849762	−	0.527167i	$-0.176746\pi$
$107$	17.5800	1.69952	0.849762	+	0.527167i	$0.176746\pi$
$108$	0	0
$109$	4.23275	0.405424	0.202712	−	0.979238i	$-0.435025\pi$
$109$	4.23275	0.405424	0.202712	+	0.979238i	$0.435025\pi$
$110$	0	0
$111$	−11.6974	−1.11027
$112$	0	0
$113$	9.07442	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$113$	9.07442	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$114$	0	0
$115$	−1.87408	−0.174759
$116$	0	0
$117$	1.86614	0.172525
$118$	0	0
$119$	−6.18883	−0.567329
$120$	0	0
$121$	−10.1226	−0.920239
$122$	0	0
$123$	−0.378801	−0.0341553
$124$	0	0
$125$	−8.69544	−0.777743
$126$	0	0
$127$	−2.02939	−0.180079	−0.0900397	−	0.995938i	$-0.528699\pi$
$127$	−2.02939	−0.180079	−0.0900397	+	0.995938i	$0.528699\pi$
$128$	0	0
$129$	8.52617	0.750687
$130$	0	0
$131$	−2.24829	−0.196434	−0.0982169	−	0.995165i	$-0.531314\pi$
$131$	−2.24829	−0.196434	−0.0982169	+	0.995165i	$0.531314\pi$
$132$	0	0
$133$	−0.579192	−0.0502224
$134$	0	0
$135$	−0.957263	−0.0823880
$136$	0	0
$137$	17.5077	1.49578	0.747892	−	0.663821i	$-0.231066\pi$
$137$	17.5077	1.49578	0.747892	+	0.663821i	$0.231066\pi$
$138$	0	0
$139$	10.4488	0.886254	0.443127	−	0.896459i	$-0.353869\pi$
$139$	10.4488	0.886254	0.443127	+	0.896459i	$0.353869\pi$
$140$	0	0
$141$	1.26703	0.106703
$142$	0	0
$143$	−1.74798	−0.146173
$144$	0	0
$145$	−0.576635	−0.0478869
$146$	0	0
$147$	6.19271	0.510766
$148$	0	0
$149$	14.6570	1.20074	0.600372	−	0.799721i	$-0.295019\pi$
$149$	14.6570	1.20074	0.600372	+	0.799721i	$0.295019\pi$
$150$	0	0
$151$	14.9495	1.21658	0.608288	−	0.793716i	$-0.291856\pi$
$151$	14.9495	1.21658	0.608288	+	0.793716i	$0.291856\pi$
$152$	0	0
$153$	6.88802	0.556863
$154$	0	0
$155$	3.27084	0.262720
$156$	0	0
$157$	−19.1037	−1.52464	−0.762320	−	0.647201i	$-0.775939\pi$
$157$	−19.1037	−1.52464	−0.762320	+	0.647201i	$0.775939\pi$
$158$	0	0
$159$	9.54479	0.756951
$160$	0	0
$161$	1.75902	0.138630
$162$	0	0
$163$	22.8963	1.79338	0.896689	−	0.442661i	$-0.145965\pi$
$163$	22.8963	1.79338	0.896689	+	0.442661i	$0.145965\pi$
$164$	0	0
$165$	0.896648	0.0698040
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.51751	−0.732116
$170$	0	0
$171$	0.644628	0.0492959
$172$	0	0
$173$	−6.22466	−0.473252	−0.236626	−	0.971601i	$-0.576042\pi$
$173$	−6.22466	−0.473252	−0.236626	+	0.971601i	$0.576042\pi$
$174$	0	0
$175$	3.66912	0.277360
$176$	0	0
$177$	1.15472	0.0867939
$178$	0	0
$179$	5.88741	0.440045	0.220023	−	0.975495i	$-0.429387\pi$
$179$	5.88741	0.440045	0.220023	+	0.975495i	$0.429387\pi$
$180$	0	0
$181$	16.5734	1.23189	0.615945	−	0.787789i	$-0.288774\pi$
$181$	16.5734	1.23189	0.615945	+	0.787789i	$0.288774\pi$
$182$	0	0
$183$	−12.4961	−0.923742
$184$	0	0
$185$	11.1975	0.823256
$186$	0	0
$187$	−6.45187	−0.471807
$188$	0	0
$189$	0.898491	0.0653556
$190$	0	0
$191$	9.02118	0.652749	0.326375	−	0.945240i	$-0.394173\pi$
$191$	9.02118	0.652749	0.326375	+	0.945240i	$0.394173\pi$
$192$	0	0
$193$	22.2488	1.60150	0.800751	−	0.598997i	$-0.204434\pi$
$193$	22.2488	1.60150	0.800751	+	0.598997i	$0.204434\pi$
$194$	0	0
$195$	−1.78639	−0.127926
$196$	0	0
$197$	16.2786	1.15980	0.579901	−	0.814687i	$-0.303091\pi$
$197$	16.2786	1.15980	0.579901	+	0.814687i	$0.303091\pi$
$198$	0	0
$199$	6.40270	0.453875	0.226938	−	0.973909i	$-0.427129\pi$
$199$	6.40270	0.453875	0.226938	+	0.973909i	$0.427129\pi$
$200$	0	0
$201$	−7.35619	−0.518866
$202$	0	0
$203$	0.541232	0.0379870
$204$	0	0
$205$	0.362612	0.0253259
$206$	0	0
$207$	−1.95775	−0.136073
$208$	0	0
$209$	−0.603810	−0.0417664
$210$	0	0
$211$	−24.7500	−1.70386	−0.851930	−	0.523656i	$-0.824568\pi$
$211$	−24.7500	−1.70386	−0.851930	+	0.523656i	$0.824568\pi$
$212$	0	0
$213$	−12.2991	−0.842723
$214$	0	0
$215$	−8.16178	−0.556629
$216$	0	0
$217$	−3.07002	−0.208407
$218$	0	0
$219$	−0.169007	−0.0114205
$220$	0	0
$221$	12.8540	0.864656
$222$	0	0
$223$	−9.03276	−0.604879	−0.302439	−	0.953169i	$-0.597801\pi$
$223$	−9.03276	−0.604879	−0.302439	+	0.953169i	$0.597801\pi$
$224$	0	0
$225$	−4.08365	−0.272243
$226$	0	0
$227$	2.59080	0.171958	0.0859788	−	0.996297i	$-0.472598\pi$
$227$	2.59080	0.171958	0.0859788	+	0.996297i	$0.472598\pi$
$228$	0	0
$229$	−16.1034	−1.06415	−0.532073	−	0.846699i	$-0.678587\pi$
$229$	−16.1034	−1.06415	−0.532073	+	0.846699i	$0.678587\pi$
$230$	0	0
$231$	−0.841599	−0.0553731
$232$	0	0
$233$	4.40995	0.288905	0.144453	−	0.989512i	$-0.453858\pi$
$233$	4.40995	0.288905	0.144453	+	0.989512i	$0.453858\pi$
$234$	0	0
$235$	−1.21288	−0.0791195
$236$	0	0
$237$	3.88101	0.252099
$238$	0	0
$239$	−21.3757	−1.38268	−0.691341	−	0.722529i	$-0.742980\pi$
$239$	−21.3757	−1.38268	−0.691341	+	0.722529i	$0.742980\pi$
$240$	0	0
$241$	−19.1359	−1.23265	−0.616326	−	0.787491i	$-0.711380\pi$
$241$	−19.1359	−1.23265	−0.616326	+	0.787491i	$0.711380\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.92805	−0.378729
$246$	0	0
$247$	1.20297	0.0765430
$248$	0	0
$249$	3.18308	0.201719
$250$	0	0
$251$	11.2928	0.712794	0.356397	−	0.934335i	$-0.384005\pi$
$251$	11.2928	0.712794	0.356397	+	0.934335i	$0.384005\pi$
$252$	0	0
$253$	1.83379	0.115289
$254$	0	0
$255$	−6.59364	−0.412910
$256$	0	0
$257$	−12.4069	−0.773922	−0.386961	−	0.922096i	$-0.626475\pi$
$257$	−12.4069	−0.773922	−0.386961	+	0.922096i	$0.626475\pi$
$258$	0	0
$259$	−10.5100	−0.653061
$260$	0	0
$261$	−0.602379	−0.0372863
$262$	0	0
$263$	−21.5868	−1.33110	−0.665550	−	0.746353i	$-0.731803\pi$
$263$	−21.5868	−1.33110	−0.665550	+	0.746353i	$0.731803\pi$
$264$	0	0
$265$	−9.13687	−0.561274
$266$	0	0
$267$	−6.13364	−0.375373
$268$	0	0
$269$	21.8764	1.33383	0.666913	−	0.745135i	$-0.267615\pi$
$269$	21.8764	1.33383	0.666913	+	0.745135i	$0.267615\pi$
$270$	0	0
$271$	18.0512	1.09653	0.548265	−	0.836304i	$-0.315288\pi$
$271$	18.0512	1.09653	0.548265	+	0.836304i	$0.315288\pi$
$272$	0	0
$273$	1.67671	0.101479
$274$	0	0
$275$	3.82507	0.230660
$276$	0	0
$277$	30.0096	1.80310	0.901552	−	0.432670i	$-0.142429\pi$
$277$	30.0096	1.80310	0.901552	+	0.432670i	$0.142429\pi$
$278$	0	0
$279$	3.41687	0.204562
$280$	0	0
$281$	12.5473	0.748508	0.374254	−	0.927326i	$-0.377899\pi$
$281$	12.5473	0.748508	0.374254	+	0.927326i	$0.377899\pi$
$282$	0	0
$283$	4.41753	0.262595	0.131297	−	0.991343i	$-0.458086\pi$
$283$	4.41753	0.262595	0.131297	+	0.991343i	$0.458086\pi$
$284$	0	0
$285$	−0.617078	−0.0365525
$286$	0	0
$287$	−0.340349	−0.0200902
$288$	0	0
$289$	30.4448	1.79087
$290$	0	0
$291$	−6.69554	−0.392499
$292$	0	0
$293$	28.2336	1.64942	0.824712	−	0.565553i	$-0.191337\pi$
$293$	28.2336	1.64942	0.824712	+	0.565553i	$0.191337\pi$
$294$	0	0
$295$	−1.10537	−0.0643570
$296$	0	0
$297$	0.936680	0.0543517
$298$	0	0
$299$	−3.65344	−0.211284
$300$	0	0
$301$	7.66069	0.441555
$302$	0	0
$303$	1.63976	0.0942018
$304$	0	0
$305$	11.9621	0.684948
$306$	0	0
$307$	−12.9138	−0.737028	−0.368514	−	0.929622i	$-0.620133\pi$
$307$	−12.9138	−0.737028	−0.368514	+	0.929622i	$0.620133\pi$
$308$	0	0
$309$	15.3568	0.873617
$310$	0	0
$311$	22.6980	1.28708	0.643542	−	0.765411i	$-0.277464\pi$
$311$	22.6980	1.28708	0.643542	+	0.765411i	$0.277464\pi$
$312$	0	0
$313$	−9.71914	−0.549358	−0.274679	−	0.961536i	$-0.588572\pi$
$313$	−9.71914	−0.549358	−0.274679	+	0.961536i	$0.588572\pi$
$314$	0	0
$315$	−0.860092	−0.0484607
$316$	0	0
$317$	20.4152	1.14663	0.573315	−	0.819335i	$-0.305657\pi$
$317$	20.4152	1.14663	0.573315	+	0.819335i	$0.305657\pi$
$318$	0	0
$319$	0.564236	0.0315911
$320$	0	0
$321$	−17.5800	−0.981220
$322$	0	0
$323$	4.44021	0.247060
$324$	0	0
$325$	−7.62067	−0.422719
$326$	0	0
$327$	−4.23275	−0.234071
$328$	0	0
$329$	1.13841	0.0627628
$330$	0	0
$331$	34.7167	1.90820	0.954102	−	0.299481i	$-0.0968136\pi$
$331$	34.7167	1.90820	0.954102	+	0.299481i	$0.0968136\pi$
$332$	0	0
$333$	11.6974	0.641014
$334$	0	0
$335$	7.04181	0.384735
$336$	0	0
$337$	15.0198	0.818179	0.409090	−	0.912494i	$-0.365846\pi$
$337$	15.0198	0.818179	0.409090	+	0.912494i	$0.365846\pi$
$338$	0	0
$339$	−9.07442	−0.492855
$340$	0	0
$341$	−3.20051	−0.173317
$342$	0	0
$343$	11.8535	0.640031
$344$	0	0
$345$	1.87408	0.100897
$346$	0	0
$347$	−34.1773	−1.83473	−0.917367	−	0.398041i	$-0.869690\pi$
$347$	−34.1773	−1.83473	−0.917367	+	0.398041i	$0.869690\pi$
$348$	0	0
$349$	−22.2400	−1.19048	−0.595240	−	0.803548i	$-0.702943\pi$
$349$	−22.2400	−1.19048	−0.595240	+	0.803548i	$0.702943\pi$
$350$	0	0
$351$	−1.86614	−0.0996074
$352$	0	0
$353$	20.8093	1.10757	0.553783	−	0.832661i	$-0.313184\pi$
$353$	20.8093	1.10757	0.553783	+	0.832661i	$0.313184\pi$
$354$	0	0
$355$	11.7735	0.624873
$356$	0	0
$357$	6.18883	0.327547
$358$	0	0
$359$	−2.72748	−0.143951	−0.0719755	−	0.997406i	$-0.522930\pi$
$359$	−2.72748	−0.143951	−0.0719755	+	0.997406i	$0.522930\pi$
$360$	0	0
$361$	−18.5845	−0.978129
$362$	0	0
$363$	10.1226	0.531300
$364$	0	0
$365$	0.161784	0.00846818
$366$	0	0
$367$	−8.20332	−0.428210	−0.214105	−	0.976811i	$-0.568683\pi$
$367$	−8.20332	−0.428210	−0.214105	+	0.976811i	$0.568683\pi$
$368$	0	0
$369$	0.378801	0.0197196
$370$	0	0
$371$	8.57591	0.445239
$372$	0	0
$373$	−3.50902	−0.181690	−0.0908451	−	0.995865i	$-0.528957\pi$
$373$	−3.50902	−0.181690	−0.0908451	+	0.995865i	$0.528957\pi$
$374$	0	0
$375$	8.69544	0.449030
$376$	0	0
$377$	−1.12413	−0.0578954
$378$	0	0
$379$	33.9160	1.74215	0.871075	−	0.491151i	$-0.163424\pi$
$379$	33.9160	1.74215	0.871075	+	0.491151i	$0.163424\pi$
$380$	0	0
$381$	2.02939	0.103969
$382$	0	0
$383$	22.6926	1.15954	0.579768	−	0.814782i	$-0.303143\pi$
$383$	22.6926	1.15954	0.579768	+	0.814782i	$0.303143\pi$
$384$	0	0
$385$	0.805631	0.0410587
$386$	0	0
$387$	−8.52617	−0.433409
$388$	0	0
$389$	1.72176	0.0872967	0.0436483	−	0.999047i	$-0.486102\pi$
$389$	1.72176	0.0872967	0.0436483	+	0.999047i	$0.486102\pi$
$390$	0	0
$391$	−13.4850	−0.681967
$392$	0	0
$393$	2.24829	0.113411
$394$	0	0
$395$	−3.71514	−0.186929
$396$	0	0
$397$	−8.23927	−0.413517	−0.206759	−	0.978392i	$-0.566291\pi$
$397$	−8.23927	−0.413517	−0.206759	+	0.978392i	$0.566291\pi$
$398$	0	0
$399$	0.579192	0.0289959
$400$	0	0
$401$	16.2106	0.809517	0.404759	−	0.914424i	$-0.367356\pi$
$401$	16.2106	0.809517	0.404759	+	0.914424i	$0.367356\pi$
$402$	0	0
$403$	6.37636	0.317629
$404$	0	0
$405$	0.957263	0.0475668
$406$	0	0
$407$	−10.9567	−0.543105
$408$	0	0
$409$	−5.21408	−0.257819	−0.128910	−	0.991656i	$-0.541148\pi$
$409$	−5.21408	−0.257819	−0.128910	+	0.991656i	$0.541148\pi$
$410$	0	0
$411$	−17.5077	−0.863591
$412$	0	0
$413$	1.03750	0.0510522
$414$	0	0
$415$	−3.04704	−0.149573
$416$	0	0
$417$	−10.4488	−0.511679
$418$	0	0
$419$	−29.4092	−1.43673	−0.718367	−	0.695664i	$-0.755110\pi$
$419$	−29.4092	−1.43673	−0.718367	+	0.695664i	$0.755110\pi$
$420$	0	0
$421$	−6.96349	−0.339380	−0.169690	−	0.985497i	$-0.554277\pi$
$421$	−6.96349	−0.339380	−0.169690	+	0.985497i	$0.554277\pi$
$422$	0	0
$423$	−1.26703	−0.0616050
$424$	0	0
$425$	−28.1282	−1.36442
$426$	0	0
$427$	−11.2277	−0.543346
$428$	0	0
$429$	1.74798	0.0843932
$430$	0	0
$431$	15.9755	0.769513	0.384756	−	0.923018i	$-0.374285\pi$
$431$	15.9755	0.769513	0.384756	+	0.923018i	$0.374285\pi$
$432$	0	0
$433$	17.0269	0.818260	0.409130	−	0.912476i	$-0.365832\pi$
$433$	17.0269	0.818260	0.409130	+	0.912476i	$0.365832\pi$
$434$	0	0
$435$	0.576635	0.0276475
$436$	0	0
$437$	−1.26202	−0.0603706
$438$	0	0
$439$	−4.69318	−0.223993	−0.111997	−	0.993709i	$-0.535725\pi$
$439$	−4.69318	−0.223993	−0.111997	+	0.993709i	$0.535725\pi$
$440$	0	0
$441$	−6.19271	−0.294891
$442$	0	0
$443$	16.9535	0.805485	0.402742	−	0.915313i	$-0.368057\pi$
$443$	16.9535	0.805485	0.402742	+	0.915313i	$0.368057\pi$
$444$	0	0
$445$	5.87151	0.278336
$446$	0	0
$447$	−14.6570	−0.693250
$448$	0	0
$449$	−2.99166	−0.141185	−0.0705926	−	0.997505i	$-0.522489\pi$
$449$	−2.99166	−0.141185	−0.0705926	+	0.997505i	$0.522489\pi$
$450$	0	0
$451$	−0.354815	−0.0167076
$452$	0	0
$453$	−14.9495	−0.702391
$454$	0	0
$455$	−1.60506	−0.0752461
$456$	0	0
$457$	−17.8935	−0.837023	−0.418512	−	0.908211i	$-0.637448\pi$
$457$	−17.8935	−0.837023	−0.418512	+	0.908211i	$0.637448\pi$
$458$	0	0
$459$	−6.88802	−0.321505
$460$	0	0
$461$	11.1815	0.520775	0.260388	−	0.965504i	$-0.416150\pi$
$461$	11.1815	0.520775	0.260388	+	0.965504i	$0.416150\pi$
$462$	0	0
$463$	−21.5593	−1.00195	−0.500974	−	0.865463i	$-0.667025\pi$
$463$	−21.5593	−1.00195	−0.500974	+	0.865463i	$0.667025\pi$
$464$	0	0
$465$	−3.27084	−0.151681
$466$	0	0
$467$	40.8278	1.88929	0.944643	−	0.328099i	$-0.106408\pi$
$467$	40.8278	1.88929	0.944643	+	0.328099i	$0.106408\pi$
$468$	0	0
$469$	−6.60948	−0.305197
$470$	0	0
$471$	19.1037	0.880251
$472$	0	0
$473$	7.98629	0.367210
$474$	0	0
$475$	−2.63243	−0.120784
$476$	0	0
$477$	−9.54479	−0.437026
$478$	0	0
$479$	32.5220	1.48597	0.742984	−	0.669309i	$-0.233410\pi$
$479$	32.5220	1.48597	0.742984	+	0.669309i	$0.233410\pi$
$480$	0	0
$481$	21.8290	0.995318
$482$	0	0
$483$	−1.75902	−0.0800383
$484$	0	0
$485$	6.40939	0.291035
$486$	0	0
$487$	−25.6360	−1.16168	−0.580839	−	0.814019i	$-0.697275\pi$
$487$	−25.6360	−1.16168	−0.580839	+	0.814019i	$0.697275\pi$
$488$	0	0
$489$	−22.8963	−1.03541
$490$	0	0
$491$	26.7553	1.20745	0.603724	−	0.797193i	$-0.293683\pi$
$491$	26.7553	1.20745	0.603724	+	0.797193i	$0.293683\pi$
$492$	0	0
$493$	−4.14920	−0.186870
$494$	0	0
$495$	−0.896648	−0.0403013
$496$	0	0
$497$	−11.0507	−0.495690
$498$	0	0
$499$	−24.6712	−1.10444	−0.552218	−	0.833700i	$-0.686218\pi$
$499$	−24.6712	−1.10444	−0.552218	+	0.833700i	$0.686218\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−22.6975	−1.01203	−0.506017	−	0.862524i	$-0.668883\pi$
$503$	−22.6975	−1.01203	−0.506017	+	0.862524i	$0.668883\pi$
$504$	0	0
$505$	−1.56968	−0.0698500
$506$	0	0
$507$	9.51751	0.422687
$508$	0	0
$509$	−12.3937	−0.549341	−0.274670	−	0.961538i	$-0.588569\pi$
$509$	−12.3937	−0.549341	−0.274670	+	0.961538i	$0.588569\pi$
$510$	0	0
$511$	−0.151852	−0.00671752
$512$	0	0
$513$	−0.644628	−0.0284610
$514$	0	0
$515$	−14.7005	−0.647781
$516$	0	0
$517$	1.18680	0.0521954
$518$	0	0
$519$	6.22466	0.273232
$520$	0	0
$521$	−17.6134	−0.771657	−0.385828	−	0.922571i	$-0.626084\pi$
$521$	−17.6134	−0.771657	−0.385828	+	0.922571i	$0.626084\pi$
$522$	0	0
$523$	−40.4400	−1.76832	−0.884159	−	0.467186i	$-0.845268\pi$
$523$	−40.4400	−1.76832	−0.884159	+	0.467186i	$0.845268\pi$
$524$	0	0
$525$	−3.66912	−0.160134
$526$	0	0
$527$	23.5354	1.02522
$528$	0	0
$529$	−19.1672	−0.833357
$530$	0	0
$531$	−1.15472	−0.0501105
$532$	0	0
$533$	0.706897	0.0306191
$534$	0	0
$535$	16.8287	0.727567
$536$	0	0
$537$	−5.88741	−0.254060
$538$	0	0
$539$	5.80059	0.249849
$540$	0	0
$541$	−23.1829	−0.996712	−0.498356	−	0.866973i	$-0.666063\pi$
$541$	−23.1829	−0.996712	−0.498356	+	0.866973i	$0.666063\pi$
$542$	0	0
$543$	−16.5734	−0.711232
$544$	0	0
$545$	4.05185	0.173562
$546$	0	0
$547$	38.5664	1.64898	0.824490	−	0.565876i	$-0.191462\pi$
$547$	38.5664	1.64898	0.824490	+	0.565876i	$0.191462\pi$
$548$	0	0
$549$	12.4961	0.533323
$550$	0	0
$551$	−0.388310	−0.0165426
$552$	0	0
$553$	3.48705	0.148285
$554$	0	0
$555$	−11.1975	−0.475307
$556$	0	0
$557$	39.5743	1.67682	0.838408	−	0.545042i	$-0.183486\pi$
$557$	39.5743	1.67682	0.838408	+	0.545042i	$0.183486\pi$
$558$	0	0
$559$	−15.9111	−0.672966
$560$	0	0
$561$	6.45187	0.272398
$562$	0	0
$563$	−6.35440	−0.267806	−0.133903	−	0.990994i	$-0.542751\pi$
$563$	−6.35440	−0.267806	−0.133903	+	0.990994i	$0.542751\pi$
$564$	0	0
$565$	8.68660	0.365448
$566$	0	0
$567$	−0.898491	−0.0377331
$568$	0	0
$569$	−25.9936	−1.08971	−0.544854	−	0.838531i	$-0.683415\pi$
$569$	−25.9936	−1.08971	−0.544854	+	0.838531i	$0.683415\pi$
$570$	0	0
$571$	−23.3120	−0.975577	−0.487788	−	0.872962i	$-0.662196\pi$
$571$	−23.3120	−0.975577	−0.487788	+	0.872962i	$0.662196\pi$
$572$	0	0
$573$	−9.02118	−0.376865
$574$	0	0
$575$	7.99477	0.333405
$576$	0	0
$577$	−13.5360	−0.563513	−0.281756	−	0.959486i	$-0.590917\pi$
$577$	−13.5360	−0.563513	−0.281756	+	0.959486i	$0.590917\pi$
$578$	0	0
$579$	−22.2488	−0.924628
$580$	0	0
$581$	2.85997	0.118651
$582$	0	0
$583$	8.94041	0.370274
$584$	0	0
$585$	1.78639	0.0738581
$586$	0	0
$587$	0.367276	0.0151591	0.00757955	−	0.999971i	$-0.497587\pi$
$587$	0.367276	0.0151591	0.00757955	+	0.999971i	$0.497587\pi$
$588$	0	0
$589$	2.20261	0.0907568
$590$	0	0
$591$	−16.2786	−0.669612
$592$	0	0
$593$	39.9010	1.63854	0.819270	−	0.573409i	$-0.194379\pi$
$593$	39.9010	1.63854	0.819270	+	0.573409i	$0.194379\pi$
$594$	0	0
$595$	−5.92433	−0.242874
$596$	0	0
$597$	−6.40270	−0.262045
$598$	0	0
$599$	−25.2610	−1.03214	−0.516068	−	0.856548i	$-0.672605\pi$
$599$	−25.2610	−1.03214	−0.516068	+	0.856548i	$0.672605\pi$
$600$	0	0
$601$	−4.06751	−0.165917	−0.0829586	−	0.996553i	$-0.526437\pi$
$601$	−4.06751	−0.165917	−0.0829586	+	0.996553i	$0.526437\pi$
$602$	0	0
$603$	7.35619	0.299567
$604$	0	0
$605$	−9.69002	−0.393955
$606$	0	0
$607$	−21.4937	−0.872403	−0.436202	−	0.899849i	$-0.643677\pi$
$607$	−21.4937	−0.872403	−0.436202	+	0.899849i	$0.643677\pi$
$608$	0	0
$609$	−0.541232	−0.0219318
$610$	0	0
$611$	−2.36446	−0.0956557
$612$	0	0
$613$	−26.4260	−1.06734	−0.533668	−	0.845694i	$-0.679187\pi$
$613$	−26.4260	−1.06734	−0.533668	+	0.845694i	$0.679187\pi$
$614$	0	0
$615$	−0.362612	−0.0146219
$616$	0	0
$617$	11.4704	0.461781	0.230890	−	0.972980i	$-0.425836\pi$
$617$	11.4704	0.461781	0.230890	+	0.972980i	$0.425836\pi$
$618$	0	0
$619$	−1.13826	−0.0457507	−0.0228753	−	0.999738i	$-0.507282\pi$
$619$	−1.13826	−0.0457507	−0.0228753	+	0.999738i	$0.507282\pi$
$620$	0	0
$621$	1.95775	0.0785618
$622$	0	0
$623$	−5.51103	−0.220795
$624$	0	0
$625$	12.0944	0.483777
$626$	0	0
$627$	0.603810	0.0241138
$628$	0	0
$629$	80.5720	3.21261
$630$	0	0
$631$	−38.2823	−1.52400	−0.761998	−	0.647580i	$-0.775781\pi$
$631$	−38.2823	−1.52400	−0.761998	+	0.647580i	$0.775781\pi$
$632$	0	0
$633$	24.7500	0.983723
$634$	0	0
$635$	−1.94266	−0.0770921
$636$	0	0
$637$	−11.5565	−0.457885
$638$	0	0
$639$	12.2991	0.486546
$640$	0	0
$641$	0.729527	0.0288146	0.0144073	−	0.999896i	$-0.495414\pi$
$641$	0.729527	0.0288146	0.0144073	+	0.999896i	$0.495414\pi$
$642$	0	0
$643$	−5.10557	−0.201344	−0.100672	−	0.994920i	$-0.532099\pi$
$643$	−5.10557	−0.201344	−0.100672	+	0.994920i	$0.532099\pi$
$644$	0	0
$645$	8.16178	0.321370
$646$	0	0
$647$	−4.04426	−0.158996	−0.0794981	−	0.996835i	$-0.525332\pi$
$647$	−4.04426	−0.158996	−0.0794981	+	0.996835i	$0.525332\pi$
$648$	0	0
$649$	1.08160	0.0424565
$650$	0	0
$651$	3.07002	0.120324
$652$	0	0
$653$	0.252341	0.00987485	0.00493743	−	0.999988i	$-0.498428\pi$
$653$	0.252341	0.00987485	0.00493743	+	0.999988i	$0.498428\pi$
$654$	0	0
$655$	−2.15220	−0.0840935
$656$	0	0
$657$	0.169007	0.00659361
$658$	0	0
$659$	19.7130	0.767908	0.383954	−	0.923352i	$-0.374562\pi$
$659$	19.7130	0.767908	0.383954	+	0.923352i	$0.374562\pi$
$660$	0	0
$661$	−33.3046	−1.29540	−0.647699	−	0.761896i	$-0.724268\pi$
$661$	−33.3046	−1.29540	−0.647699	+	0.761896i	$0.724268\pi$
$662$	0	0
$663$	−12.8540	−0.499209
$664$	0	0
$665$	−0.554439	−0.0215002
$666$	0	0
$667$	1.17931	0.0456630
$668$	0	0
$669$	9.03276	0.349227
$670$	0	0
$671$	−11.7049	−0.451862
$672$	0	0
$673$	−24.1873	−0.932353	−0.466176	−	0.884692i	$-0.654369\pi$
$673$	−24.1873	−0.932353	−0.466176	+	0.884692i	$0.654369\pi$
$674$	0	0
$675$	4.08365	0.157180
$676$	0	0
$677$	10.2383	0.393489	0.196744	−	0.980455i	$-0.436963\pi$
$677$	10.2383	0.393489	0.196744	+	0.980455i	$0.436963\pi$
$678$	0	0
$679$	−6.01588	−0.230868
$680$	0	0
$681$	−2.59080	−0.0992798
$682$	0	0
$683$	−25.9050	−0.991226	−0.495613	−	0.868543i	$-0.665057\pi$
$683$	−25.9050	−0.991226	−0.495613	+	0.868543i	$0.665057\pi$
$684$	0	0
$685$	16.7595	0.640346
$686$	0	0
$687$	16.1034	0.614385
$688$	0	0
$689$	−17.8120	−0.678581
$690$	0	0
$691$	28.3962	1.08024	0.540121	−	0.841588i	$-0.318379\pi$
$691$	28.3962	1.08024	0.540121	+	0.841588i	$0.318379\pi$
$692$	0	0
$693$	0.841599	0.0319697
$694$	0	0
$695$	10.0022	0.379406
$696$	0	0
$697$	2.60919	0.0988301
$698$	0	0
$699$	−4.40995	−0.166800
$700$	0	0
$701$	−50.6482	−1.91295	−0.956477	−	0.291806i	$-0.905744\pi$
$701$	−50.6482	−1.91295	−0.956477	+	0.291806i	$0.905744\pi$
$702$	0	0
$703$	7.54047	0.284394
$704$	0	0
$705$	1.21288	0.0456797
$706$	0	0
$707$	1.47331	0.0554096
$708$	0	0
$709$	−44.0789	−1.65542	−0.827710	−	0.561157i	$-0.810357\pi$
$709$	−44.0789	−1.65542	−0.827710	+	0.561157i	$0.810357\pi$
$710$	0	0
$711$	−3.88101	−0.145549
$712$	0	0
$713$	−6.68937	−0.250519
$714$	0	0
$715$	−1.67327	−0.0625769
$716$	0	0
$717$	21.3757	0.798291
$718$	0	0
$719$	−14.6295	−0.545587	−0.272793	−	0.962073i	$-0.587948\pi$
$719$	−14.6295	−0.545587	−0.272793	+	0.962073i	$0.587948\pi$
$720$	0	0
$721$	13.7979	0.513862
$722$	0	0
$723$	19.1359	0.711672
$724$	0	0
$725$	2.45990	0.0913585
$726$	0	0
$727$	−52.4926	−1.94684	−0.973421	−	0.229023i	$-0.926447\pi$
$727$	−52.4926	−1.94684	−0.973421	+	0.229023i	$0.926447\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−58.7284	−2.17215
$732$	0	0
$733$	12.5837	0.464791	0.232396	−	0.972621i	$-0.425344\pi$
$733$	12.5837	0.464791	0.232396	+	0.972621i	$0.425344\pi$
$734$	0	0
$735$	5.92805	0.218660
$736$	0	0
$737$	−6.89040	−0.253811
$738$	0	0
$739$	−0.539612	−0.0198500	−0.00992498	−	0.999951i	$-0.503159\pi$
$739$	−0.539612	−0.0198500	−0.00992498	+	0.999951i	$0.503159\pi$
$740$	0	0
$741$	−1.20297	−0.0441921
$742$	0	0
$743$	−11.9778	−0.439422	−0.219711	−	0.975565i	$-0.570511\pi$
$743$	−11.9778	−0.439422	−0.219711	+	0.975565i	$0.570511\pi$
$744$	0	0
$745$	14.0306	0.514040
$746$	0	0
$747$	−3.18308	−0.116463
$748$	0	0
$749$	−15.7955	−0.577154
$750$	0	0
$751$	−31.2764	−1.14129	−0.570647	−	0.821196i	$-0.693307\pi$
$751$	−31.2764	−1.14129	−0.570647	+	0.821196i	$0.693307\pi$
$752$	0	0
$753$	−11.2928	−0.411532
$754$	0	0
$755$	14.3106	0.520818
$756$	0	0
$757$	−13.8376	−0.502935	−0.251467	−	0.967866i	$-0.580913\pi$
$757$	−13.8376	−0.502935	−0.251467	+	0.967866i	$0.580913\pi$
$758$	0	0
$759$	−1.83379	−0.0665622
$760$	0	0
$761$	−14.8220	−0.537297	−0.268649	−	0.963238i	$-0.586577\pi$
$761$	−14.8220	−0.537297	−0.268649	+	0.963238i	$0.586577\pi$
$762$	0	0
$763$	−3.80309	−0.137681
$764$	0	0
$765$	6.59364	0.238394
$766$	0	0
$767$	−2.15487	−0.0778078
$768$	0	0
$769$	−30.8191	−1.11136	−0.555682	−	0.831395i	$-0.687543\pi$
$769$	−30.8191	−1.11136	−0.555682	+	0.831395i	$0.687543\pi$
$770$	0	0
$771$	12.4069	0.446824
$772$	0	0
$773$	9.42958	0.339159	0.169579	−	0.985517i	$-0.445759\pi$
$773$	9.42958	0.339159	0.169579	+	0.985517i	$0.445759\pi$
$774$	0	0
$775$	−13.9533	−0.501217
$776$	0	0
$777$	10.5100	0.377045
$778$	0	0
$779$	0.244186	0.00874886
$780$	0	0
$781$	−11.5204	−0.412231
$782$	0	0
$783$	0.602379	0.0215273
$784$	0	0
$785$	−18.2872	−0.652699
$786$	0	0
$787$	24.6894	0.880081	0.440041	−	0.897978i	$-0.354964\pi$
$787$	24.6894	0.880081	0.440041	+	0.897978i	$0.354964\pi$
$788$	0	0
$789$	21.5868	0.768511
$790$	0	0
$791$	−8.15329	−0.289897
$792$	0	0
$793$	23.3196	0.828103
$794$	0	0
$795$	9.13687	0.324051
$796$	0	0
$797$	−16.2017	−0.573893	−0.286947	−	0.957947i	$-0.592640\pi$
$797$	−16.2017	−0.573893	−0.286947	+	0.957947i	$0.592640\pi$
$798$	0	0
$799$	−8.72731	−0.308750
$800$	0	0
$801$	6.13364	0.216722
$802$	0	0
$803$	−0.158306	−0.00558649
$804$	0	0
$805$	1.68385	0.0593478
$806$	0	0
$807$	−21.8764	−0.770085
$808$	0	0
$809$	−5.73547	−0.201648	−0.100824	−	0.994904i	$-0.532148\pi$
$809$	−5.73547	−0.201648	−0.100824	+	0.994904i	$0.532148\pi$
$810$	0	0
$811$	36.0299	1.26518	0.632590	−	0.774487i	$-0.281992\pi$
$811$	36.0299	1.26518	0.632590	+	0.774487i	$0.281992\pi$
$812$	0	0
$813$	−18.0512	−0.633082
$814$	0	0
$815$	21.9178	0.767747
$816$	0	0
$817$	−5.49620	−0.192288
$818$	0	0
$819$	−1.67671	−0.0585891
$820$	0	0
$821$	−12.5282	−0.437238	−0.218619	−	0.975810i	$-0.570155\pi$
$821$	−12.5282	−0.437238	−0.218619	+	0.975810i	$0.570155\pi$
$822$	0	0
$823$	43.0141	1.49938	0.749689	−	0.661791i	$-0.230203\pi$
$823$	43.0141	1.49938	0.749689	+	0.661791i	$0.230203\pi$
$824$	0	0
$825$	−3.82507	−0.133172
$826$	0	0
$827$	−20.6080	−0.716612	−0.358306	−	0.933604i	$-0.616645\pi$
$827$	−20.6080	−0.716612	−0.358306	+	0.933604i	$0.616645\pi$
$828$	0	0
$829$	−29.3130	−1.01808	−0.509040	−	0.860743i	$-0.670000\pi$
$829$	−29.3130	−1.01808	−0.509040	+	0.860743i	$0.670000\pi$
$830$	0	0
$831$	−30.0096	−1.04102
$832$	0	0
$833$	−42.6555	−1.47793
$834$	0	0
$835$	0.957263	0.0331274
$836$	0	0
$837$	−3.41687	−0.118104
$838$	0	0
$839$	29.9614	1.03438	0.517191	−	0.855870i	$-0.326978\pi$
$839$	29.9614	1.03438	0.517191	+	0.855870i	$0.326978\pi$
$840$	0	0
$841$	−28.6371	−0.987488
$842$	0	0
$843$	−12.5473	−0.432151
$844$	0	0
$845$	−9.11076	−0.313420
$846$	0	0
$847$	9.09510	0.312511
$848$	0	0
$849$	−4.41753	−0.151609
$850$	0	0
$851$	−22.9006	−0.785023
$852$	0	0
$853$	47.0657	1.61150	0.805750	−	0.592256i	$-0.201762\pi$
$853$	47.0657	1.61150	0.805750	+	0.592256i	$0.201762\pi$
$854$	0	0
$855$	0.617078	0.0211036
$856$	0	0
$857$	−14.7556	−0.504041	−0.252020	−	0.967722i	$-0.581095\pi$
$857$	−14.7556	−0.504041	−0.252020	+	0.967722i	$0.581095\pi$
$858$	0	0
$859$	25.5694	0.872416	0.436208	−	0.899846i	$-0.356321\pi$
$859$	25.5694	0.872416	0.436208	+	0.899846i	$0.356321\pi$
$860$	0	0
$861$	0.340349	0.0115991
$862$	0	0
$863$	−26.7717	−0.911320	−0.455660	−	0.890154i	$-0.650597\pi$
$863$	−26.7717	−0.911320	−0.455660	+	0.890154i	$0.650597\pi$
$864$	0	0
$865$	−5.95863	−0.202600
$866$	0	0
$867$	−30.4448	−1.03396
$868$	0	0
$869$	3.63526	0.123318
$870$	0	0
$871$	13.7277	0.465146
$872$	0	0
$873$	6.69554	0.226610
$874$	0	0
$875$	7.81277	0.264120
$876$	0	0
$877$	−41.9329	−1.41597	−0.707986	−	0.706226i	$-0.750396\pi$
$877$	−41.9329	−1.41597	−0.707986	+	0.706226i	$0.750396\pi$
$878$	0	0
$879$	−28.2336	−0.952295
$880$	0	0
$881$	−38.2857	−1.28988	−0.644939	−	0.764234i	$-0.723117\pi$
$881$	−38.2857	−1.28988	−0.644939	+	0.764234i	$0.723117\pi$
$882$	0	0
$883$	23.8441	0.802417	0.401209	−	0.915987i	$-0.368590\pi$
$883$	23.8441	0.802417	0.401209	+	0.915987i	$0.368590\pi$
$884$	0	0
$885$	1.10537	0.0371565
$886$	0	0
$887$	−4.51521	−0.151606	−0.0758030	−	0.997123i	$-0.524152\pi$
$887$	−4.51521	−0.151606	−0.0758030	+	0.997123i	$0.524152\pi$
$888$	0	0
$889$	1.82339	0.0611546
$890$	0	0
$891$	−0.936680	−0.0313799
$892$	0	0
$893$	−0.816762	−0.0273319
$894$	0	0
$895$	5.63579	0.188384
$896$	0	0
$897$	3.65344	0.121985
$898$	0	0
$899$	−2.05825	−0.0686464
$900$	0	0
$901$	−65.7447	−2.19027
$902$	0	0
$903$	−7.66069	−0.254932
$904$	0	0
$905$	15.8651	0.527373
$906$	0	0
$907$	23.7096	0.787265	0.393633	−	0.919268i	$-0.371218\pi$
$907$	23.7096	0.787265	0.393633	+	0.919268i	$0.371218\pi$
$908$	0	0
$909$	−1.63976	−0.0543875
$910$	0	0
$911$	−26.7056	−0.884797	−0.442399	−	0.896819i	$-0.645872\pi$
$911$	−26.7056	−0.884797	−0.442399	+	0.896819i	$0.645872\pi$
$912$	0	0
$913$	2.98152	0.0986741
$914$	0	0
$915$	−11.9621	−0.395455
$916$	0	0
$917$	2.02007	0.0667085
$918$	0	0
$919$	−49.7330	−1.64054	−0.820271	−	0.571975i	$-0.806177\pi$
$919$	−49.7330	−1.64054	−0.820271	+	0.571975i	$0.806177\pi$
$920$	0	0
$921$	12.9138	0.425524
$922$	0	0
$923$	22.9520	0.755473
$924$	0	0
$925$	−47.7681	−1.57061
$926$	0	0
$927$	−15.3568	−0.504383
$928$	0	0
$929$	50.3384	1.65155	0.825774	−	0.564001i	$-0.190739\pi$
$929$	50.3384	1.65155	0.825774	+	0.564001i	$0.190739\pi$
$930$	0	0
$931$	−3.99199	−0.130832
$932$	0	0
$933$	−22.6980	−0.743099
$934$	0	0
$935$	−6.17613	−0.201981
$936$	0	0
$937$	44.8159	1.46407	0.732036	−	0.681266i	$-0.238570\pi$
$937$	44.8159	1.46407	0.732036	+	0.681266i	$0.238570\pi$
$938$	0	0
$939$	9.71914	0.317172
$940$	0	0
$941$	4.20187	0.136977	0.0684885	−	0.997652i	$-0.478182\pi$
$941$	4.20187	0.136977	0.0684885	+	0.997652i	$0.478182\pi$
$942$	0	0
$943$	−0.741598	−0.0241498
$944$	0	0
$945$	0.860092	0.0279788
$946$	0	0
$947$	45.7156	1.48556	0.742778	−	0.669537i	$-0.233508\pi$
$947$	45.7156	1.48556	0.742778	+	0.669537i	$0.233508\pi$
$948$	0	0
$949$	0.315392	0.0102381
$950$	0	0
$951$	−20.4152	−0.662008
$952$	0	0
$953$	29.7636	0.964138	0.482069	−	0.876133i	$-0.339885\pi$
$953$	29.7636	0.964138	0.482069	+	0.876133i	$0.339885\pi$
$954$	0	0
$955$	8.63564	0.279443
$956$	0	0
$957$	−0.564236	−0.0182392
$958$	0	0
$959$	−15.7305	−0.507965
$960$	0	0
$961$	−19.3250	−0.623388
$962$	0	0
$963$	17.5800	0.566508
$964$	0	0
$965$	21.2979	0.685604
$966$	0	0
$967$	−24.3229	−0.782170	−0.391085	−	0.920355i	$-0.627900\pi$
$967$	−24.3229	−0.782170	−0.391085	+	0.920355i	$0.627900\pi$
$968$	0	0
$969$	−4.44021	−0.142640
$970$	0	0
$971$	−44.8086	−1.43798	−0.718989	−	0.695022i	$-0.755395\pi$
$971$	−44.8086	−1.43798	−0.718989	+	0.695022i	$0.755395\pi$
$972$	0	0
$973$	−9.38814	−0.300970
$974$	0	0
$975$	7.62067	0.244057
$976$	0	0
$977$	25.8444	0.826836	0.413418	−	0.910541i	$-0.364335\pi$
$977$	25.8444	0.826836	0.413418	+	0.910541i	$0.364335\pi$
$978$	0	0
$979$	−5.74526	−0.183619
$980$	0	0
$981$	4.23275	0.135141
$982$	0	0
$983$	−11.9377	−0.380754	−0.190377	−	0.981711i	$-0.560971\pi$
$983$	−11.9377	−0.380754	−0.190377	+	0.981711i	$0.560971\pi$
$984$	0	0
$985$	15.5829	0.496512
$986$	0	0
$987$	−1.13841	−0.0362361
$988$	0	0
$989$	16.6921	0.530778
$990$	0	0
$991$	12.7875	0.406209	0.203105	−	0.979157i	$-0.434897\pi$
$991$	12.7875	0.406209	0.203105	+	0.979157i	$0.434897\pi$
$992$	0	0
$993$	−34.7167	−1.10170
$994$	0	0
$995$	6.12906	0.194304
$996$	0	0
$997$	52.0763	1.64927	0.824636	−	0.565664i	$-0.191380\pi$
$997$	52.0763	1.64927	0.824636	+	0.565664i	$0.191380\pi$
$998$	0	0
$999$	−11.6974	−0.370090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.k.1.6	✓	11
4.3	odd	2		8016.2.a.be.1.6		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.6	✓	11	1.1	even	1	trivial
8016.2.a.be.1.6		11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.957263 q^{5} -0.898491 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.957263 q^{5} -0.898491 q^{7} +1.00000 q^{9} -0.936680 q^{11} +1.86614 q^{13} -0.957263 q^{15} +6.88802 q^{17} +0.644628 q^{19} +0.898491 q^{21} -1.95775 q^{23} -4.08365 q^{25} -1.00000 q^{27} -0.602379 q^{29} +3.41687 q^{31} +0.936680 q^{33} -0.860092 q^{35} +11.6974 q^{37} -1.86614 q^{39} +0.378801 q^{41} -8.52617 q^{43} +0.957263 q^{45} -1.26703 q^{47} -6.19271 q^{49} -6.88802 q^{51} -9.54479 q^{53} -0.896648 q^{55} -0.644628 q^{57} -1.15472 q^{59} +12.4961 q^{61} -0.898491 q^{63} +1.78639 q^{65} +7.35619 q^{67} +1.95775 q^{69} +12.2991 q^{71} +0.169007 q^{73} +4.08365 q^{75} +0.841599 q^{77} -3.88101 q^{79} +1.00000 q^{81} -3.18308 q^{83} +6.59364 q^{85} +0.602379 q^{87} +6.13364 q^{89} -1.67671 q^{91} -3.41687 q^{93} +0.617078 q^{95} +6.69554 q^{97} -0.936680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9} - q^{11} + 10 q^{13} - 10 q^{15} + 17 q^{17} + 2 q^{19} + q^{21} - 3 q^{23} + 21 q^{25} - 11 q^{27} + 17 q^{29} - 15 q^{31} + q^{33} + 11 q^{35} + 4 q^{37} - 10 q^{39} + 16 q^{41} + 10 q^{43} + 10 q^{45} - 16 q^{47} + 22 q^{49} - 17 q^{51} + 42 q^{53} - 5 q^{55} - 2 q^{57} - 2 q^{59} + 12 q^{61} - q^{63} + 10 q^{65} - q^{67} + 3 q^{69} - 9 q^{71} + 24 q^{73} - 21 q^{75} + 22 q^{77} - 30 q^{79} + 11 q^{81} + 16 q^{83} + 25 q^{85} - 17 q^{87} + 37 q^{89} + q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) \) 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9 - q^11 + 10 * q^13 - 10 * q^15 + 17 * q^17 + 2 * q^19 + q^21 - 3 * q^23 + 21 * q^25 - 11 * q^27 + 17 * q^29 - 15 * q^31 + q^33 + 11 * q^35 + 4 * q^37 - 10 * q^39 + 16 * q^41 + 10 * q^43 + 10 * q^45 - 16 * q^47 + 22 * q^49 - 17 * q^51 + 42 * q^53 - 5 * q^55 - 2 * q^57 - 2 * q^59 + 12 * q^61 - q^63 + 10 * q^65 - q^67 + 3 * q^69 - 9 * q^71 + 24 * q^73 - 21 * q^75 + 22 * q^77 - 30 * q^79 + 11 * q^81 + 16 * q^83 + 25 * q^85 - 17 * q^87 + 37 * q^89 + q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97 - q^99

Introduction

L-functions

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Groups

Database

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