LMFDB - Embedded newform 4008.2.a.l.1.4

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.14709$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.14709 q^{5} -3.32442 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.14709 q^{5} -3.32442 q^{7} +1.00000 q^{9} -2.04384 q^{11} +5.67233 q^{13} -1.14709 q^{15} -4.44387 q^{17} +4.57483 q^{19} -3.32442 q^{21} -8.33021 q^{23} -3.68419 q^{25} +1.00000 q^{27} -6.09591 q^{29} +10.8446 q^{31} -2.04384 q^{33} +3.81340 q^{35} +3.37692 q^{37} +5.67233 q^{39} +3.28441 q^{41} +10.1171 q^{43} -1.14709 q^{45} +7.22397 q^{47} +4.05178 q^{49} -4.44387 q^{51} +8.07461 q^{53} +2.34446 q^{55} +4.57483 q^{57} -0.491745 q^{59} -1.93432 q^{61} -3.32442 q^{63} -6.50665 q^{65} +5.76932 q^{67} -8.33021 q^{69} +8.42446 q^{71} -14.6540 q^{73} -3.68419 q^{75} +6.79458 q^{77} +0.640102 q^{79} +1.00000 q^{81} +12.9891 q^{83} +5.09750 q^{85} -6.09591 q^{87} -13.8518 q^{89} -18.8572 q^{91} +10.8446 q^{93} -5.24773 q^{95} -14.0653 q^{97} -2.04384 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * 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
$5$	−1.14709	−0.512993	−0.256496	−	0.966545i	$-0.582568\pi$
$5$	−1.14709	−0.512993	−0.256496	+	0.966545i	$0.582568\pi$
$6$	0	0
$7$	−3.32442	−1.25651	−0.628257	−	0.778006i	$-0.716231\pi$
$7$	−3.32442	−1.25651	−0.628257	+	0.778006i	$0.716231\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.04384	−0.616240	−0.308120	−	0.951347i	$-0.599700\pi$
$11$	−2.04384	−0.616240	−0.308120	+	0.951347i	$0.599700\pi$
$12$	0	0
$13$	5.67233	1.57322	0.786610	−	0.617450i	$-0.211834\pi$
$13$	5.67233	1.57322	0.786610	+	0.617450i	$0.211834\pi$
$14$	0	0
$15$	−1.14709	−0.296176
$16$	0	0
$17$	−4.44387	−1.07780	−0.538898	−	0.842371i	$-0.681159\pi$
$17$	−4.44387	−1.07780	−0.538898	+	0.842371i	$0.681159\pi$
$18$	0	0
$19$	4.57483	1.04954	0.524769	−	0.851245i	$-0.324152\pi$
$19$	4.57483	1.04954	0.524769	+	0.851245i	$0.324152\pi$
$20$	0	0
$21$	−3.32442	−0.725448
$22$	0	0
$23$	−8.33021	−1.73697	−0.868484	−	0.495717i	$-0.834905\pi$
$23$	−8.33021	−1.73697	−0.868484	+	0.495717i	$0.834905\pi$
$24$	0	0
$25$	−3.68419	−0.736839
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.09591	−1.13198	−0.565991	−	0.824411i	$-0.691506\pi$
$29$	−6.09591	−1.13198	−0.565991	+	0.824411i	$0.691506\pi$
$30$	0	0
$31$	10.8446	1.94774	0.973871	−	0.227100i	$-0.0729244\pi$
$31$	10.8446	1.94774	0.973871	+	0.227100i	$0.0729244\pi$
$32$	0	0
$33$	−2.04384	−0.355786
$34$	0	0
$35$	3.81340	0.644582
$36$	0	0
$37$	3.37692	0.555162	0.277581	−	0.960702i	$-0.410467\pi$
$37$	3.37692	0.555162	0.277581	+	0.960702i	$0.410467\pi$
$38$	0	0
$39$	5.67233	0.908299
$40$	0	0
$41$	3.28441	0.512939	0.256469	−	0.966552i	$-0.417441\pi$
$41$	3.28441	0.512939	0.256469	+	0.966552i	$0.417441\pi$
$42$	0	0
$43$	10.1171	1.54285	0.771423	−	0.636322i	$-0.219545\pi$
$43$	10.1171	1.54285	0.771423	+	0.636322i	$0.219545\pi$
$44$	0	0
$45$	−1.14709	−0.170998
$46$	0	0
$47$	7.22397	1.05372	0.526862	−	0.849951i	$-0.323368\pi$
$47$	7.22397	1.05372	0.526862	+	0.849951i	$0.323368\pi$
$48$	0	0
$49$	4.05178	0.578826
$50$	0	0
$51$	−4.44387	−0.622266
$52$	0	0
$53$	8.07461	1.10913	0.554567	−	0.832139i	$-0.312884\pi$
$53$	8.07461	1.10913	0.554567	+	0.832139i	$0.312884\pi$
$54$	0	0
$55$	2.34446	0.316127
$56$	0	0
$57$	4.57483	0.605951
$58$	0	0
$59$	−0.491745	−0.0640198	−0.0320099	−	0.999488i	$-0.510191\pi$
$59$	−0.491745	−0.0640198	−0.0320099	+	0.999488i	$0.510191\pi$
$60$	0	0
$61$	−1.93432	−0.247665	−0.123832	−	0.992303i	$-0.539518\pi$
$61$	−1.93432	−0.247665	−0.123832	+	0.992303i	$0.539518\pi$
$62$	0	0
$63$	−3.32442	−0.418838
$64$	0	0
$65$	−6.50665	−0.807051
$66$	0	0
$67$	5.76932	0.704835	0.352417	−	0.935843i	$-0.385360\pi$
$67$	5.76932	0.704835	0.352417	+	0.935843i	$0.385360\pi$
$68$	0	0
$69$	−8.33021	−1.00284
$70$	0	0
$71$	8.42446	0.999799	0.499900	−	0.866083i	$-0.333370\pi$
$71$	8.42446	0.999799	0.499900	+	0.866083i	$0.333370\pi$
$72$	0	0
$73$	−14.6540	−1.71512	−0.857560	−	0.514384i	$-0.828021\pi$
$73$	−14.6540	−1.71512	−0.857560	+	0.514384i	$0.828021\pi$
$74$	0	0
$75$	−3.68419	−0.425414
$76$	0	0
$77$	6.79458	0.774314
$78$	0	0
$79$	0.640102	0.0720171	0.0360085	−	0.999351i	$-0.488536\pi$
$79$	0.640102	0.0720171	0.0360085	+	0.999351i	$0.488536\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9891	1.42574	0.712871	−	0.701296i	$-0.247395\pi$
$83$	12.9891	1.42574	0.712871	+	0.701296i	$0.247395\pi$
$84$	0	0
$85$	5.09750	0.552902
$86$	0	0
$87$	−6.09591	−0.653550
$88$	0	0
$89$	−13.8518	−1.46828	−0.734142	−	0.678996i	$-0.762415\pi$
$89$	−13.8518	−1.46828	−0.734142	+	0.678996i	$0.762415\pi$
$90$	0	0
$91$	−18.8572	−1.97677
$92$	0	0
$93$	10.8446	1.12453
$94$	0	0
$95$	−5.24773	−0.538406
$96$	0	0
$97$	−14.0653	−1.42811	−0.714056	−	0.700089i	$-0.753144\pi$
$97$	−14.0653	−1.42811	−0.714056	+	0.700089i	$0.753144\pi$
$98$	0	0
$99$	−2.04384	−0.205413
$100$	0	0
$101$	9.38072	0.933417	0.466708	−	0.884411i	$-0.345440\pi$
$101$	9.38072	0.933417	0.466708	+	0.884411i	$0.345440\pi$
$102$	0	0
$103$	13.9780	1.37729	0.688645	−	0.725099i	$-0.258206\pi$
$103$	13.9780	1.37729	0.688645	+	0.725099i	$0.258206\pi$
$104$	0	0
$105$	3.81340	0.372150
$106$	0	0
$107$	−8.32030	−0.804353	−0.402177	−	0.915562i	$-0.631746\pi$
$107$	−8.32030	−0.804353	−0.402177	+	0.915562i	$0.631746\pi$
$108$	0	0
$109$	16.2370	1.55522	0.777610	−	0.628748i	$-0.216432\pi$
$109$	16.2370	1.55522	0.777610	+	0.628748i	$0.216432\pi$
$110$	0	0
$111$	3.37692	0.320523
$112$	0	0
$113$	10.0624	0.946595	0.473297	−	0.880903i	$-0.343064\pi$
$113$	10.0624	0.946595	0.473297	+	0.880903i	$0.343064\pi$
$114$	0	0
$115$	9.55547	0.891052
$116$	0	0
$117$	5.67233	0.524407
$118$	0	0
$119$	14.7733	1.35427
$120$	0	0
$121$	−6.82273	−0.620248
$122$	0	0
$123$	3.28441	0.296145
$124$	0	0
$125$	9.96152	0.890985
$126$	0	0
$127$	−6.58874	−0.584656	−0.292328	−	0.956318i	$-0.594430\pi$
$127$	−6.58874	−0.584656	−0.292328	+	0.956318i	$0.594430\pi$
$128$	0	0
$129$	10.1171	0.890763
$130$	0	0
$131$	−3.88569	−0.339494	−0.169747	−	0.985488i	$-0.554295\pi$
$131$	−3.88569	−0.339494	−0.169747	+	0.985488i	$0.554295\pi$
$132$	0	0
$133$	−15.2087	−1.31876
$134$	0	0
$135$	−1.14709	−0.0987255
$136$	0	0
$137$	20.3679	1.74015	0.870075	−	0.492919i	$-0.164070\pi$
$137$	20.3679	1.74015	0.870075	+	0.492919i	$0.164070\pi$
$138$	0	0
$139$	14.2985	1.21278	0.606390	−	0.795168i	$-0.292617\pi$
$139$	14.2985	1.21278	0.606390	+	0.795168i	$0.292617\pi$
$140$	0	0
$141$	7.22397	0.608368
$142$	0	0
$143$	−11.5933	−0.969482
$144$	0	0
$145$	6.99254	0.580699
$146$	0	0
$147$	4.05178	0.334185
$148$	0	0
$149$	13.5233	1.10787	0.553936	−	0.832559i	$-0.313125\pi$
$149$	13.5233	1.10787	0.553936	+	0.832559i	$0.313125\pi$
$150$	0	0
$151$	2.39755	0.195110	0.0975549	−	0.995230i	$-0.468898\pi$
$151$	2.39755	0.195110	0.0975549	+	0.995230i	$0.468898\pi$
$152$	0	0
$153$	−4.44387	−0.359265
$154$	0	0
$155$	−12.4397	−0.999178
$156$	0	0
$157$	−1.40129	−0.111835	−0.0559174	−	0.998435i	$-0.517808\pi$
$157$	−1.40129	−0.111835	−0.0559174	+	0.998435i	$0.517808\pi$
$158$	0	0
$159$	8.07461	0.640358
$160$	0	0
$161$	27.6931	2.18252
$162$	0	0
$163$	13.9978	1.09639	0.548197	−	0.836350i	$-0.315315\pi$
$163$	13.9978	1.09639	0.548197	+	0.836350i	$0.315315\pi$
$164$	0	0
$165$	2.34446	0.182516
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	19.1753	1.47502
$170$	0	0
$171$	4.57483	0.349846
$172$	0	0
$173$	2.70111	0.205361	0.102681	−	0.994714i	$-0.467258\pi$
$173$	2.70111	0.205361	0.102681	+	0.994714i	$0.467258\pi$
$174$	0	0
$175$	12.2478	0.925848
$176$	0	0
$177$	−0.491745	−0.0369618
$178$	0	0
$179$	−2.99486	−0.223846	−0.111923	−	0.993717i	$-0.535701\pi$
$179$	−2.99486	−0.223846	−0.111923	+	0.993717i	$0.535701\pi$
$180$	0	0
$181$	13.5390	1.00634	0.503171	−	0.864187i	$-0.332166\pi$
$181$	13.5390	1.00634	0.503171	+	0.864187i	$0.332166\pi$
$182$	0	0
$183$	−1.93432	−0.142989
$184$	0	0
$185$	−3.87362	−0.284794
$186$	0	0
$187$	9.08254	0.664181
$188$	0	0
$189$	−3.32442	−0.241816
$190$	0	0
$191$	−18.3703	−1.32923	−0.664613	−	0.747188i	$-0.731404\pi$
$191$	−18.3703	−1.32923	−0.664613	+	0.747188i	$0.731404\pi$
$192$	0	0
$193$	−8.15030	−0.586671	−0.293336	−	0.956010i	$-0.594765\pi$
$193$	−8.15030	−0.586671	−0.293336	+	0.956010i	$0.594765\pi$
$194$	0	0
$195$	−6.50665	−0.465951
$196$	0	0
$197$	9.56487	0.681469	0.340734	−	0.940160i	$-0.389324\pi$
$197$	9.56487	0.681469	0.340734	+	0.940160i	$0.389324\pi$
$198$	0	0
$199$	−24.9138	−1.76609	−0.883045	−	0.469289i	$-0.844510\pi$
$199$	−24.9138	−1.76609	−0.883045	+	0.469289i	$0.844510\pi$
$200$	0	0
$201$	5.76932	0.406937
$202$	0	0
$203$	20.2654	1.42235
$204$	0	0
$205$	−3.76750	−0.263134
$206$	0	0
$207$	−8.33021	−0.578989
$208$	0	0
$209$	−9.35021	−0.646768
$210$	0	0
$211$	21.2970	1.46615	0.733073	−	0.680150i	$-0.238085\pi$
$211$	21.2970	1.46615	0.733073	+	0.680150i	$0.238085\pi$
$212$	0	0
$213$	8.42446	0.577234
$214$	0	0
$215$	−11.6052	−0.791469
$216$	0	0
$217$	−36.0519	−2.44737
$218$	0	0
$219$	−14.6540	−0.990225
$220$	0	0
$221$	−25.2071	−1.69561
$222$	0	0
$223$	21.5679	1.44430	0.722148	−	0.691738i	$-0.243155\pi$
$223$	21.5679	1.44430	0.722148	+	0.691738i	$0.243155\pi$
$224$	0	0
$225$	−3.68419	−0.245613
$226$	0	0
$227$	23.9038	1.58655	0.793275	−	0.608863i	$-0.208374\pi$
$227$	23.9038	1.58655	0.793275	+	0.608863i	$0.208374\pi$
$228$	0	0
$229$	5.37332	0.355079	0.177539	−	0.984114i	$-0.443186\pi$
$229$	5.37332	0.355079	0.177539	+	0.984114i	$0.443186\pi$
$230$	0	0
$231$	6.79458	0.447050
$232$	0	0
$233$	−22.0472	−1.44436	−0.722179	−	0.691706i	$-0.756859\pi$
$233$	−22.0472	−1.44436	−0.722179	+	0.691706i	$0.756859\pi$
$234$	0	0
$235$	−8.28652	−0.540553
$236$	0	0
$237$	0.640102	0.0415791
$238$	0	0
$239$	−11.1537	−0.721473	−0.360737	−	0.932668i	$-0.617475\pi$
$239$	−11.1537	−0.721473	−0.360737	+	0.932668i	$0.617475\pi$
$240$	0	0
$241$	−6.82085	−0.439370	−0.219685	−	0.975571i	$-0.570503\pi$
$241$	−6.82085	−0.439370	−0.219685	+	0.975571i	$0.570503\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.64775	−0.296934
$246$	0	0
$247$	25.9500	1.65116
$248$	0	0
$249$	12.9891	0.823152
$250$	0	0
$251$	−14.3413	−0.905216	−0.452608	−	0.891710i	$-0.649506\pi$
$251$	−14.3413	−0.905216	−0.452608	+	0.891710i	$0.649506\pi$
$252$	0	0
$253$	17.0256	1.07039
$254$	0	0
$255$	5.09750	0.319218
$256$	0	0
$257$	−6.18565	−0.385850	−0.192925	−	0.981213i	$-0.561797\pi$
$257$	−6.18565	−0.385850	−0.192925	+	0.981213i	$0.561797\pi$
$258$	0	0
$259$	−11.2263	−0.697569
$260$	0	0
$261$	−6.09591	−0.377327
$262$	0	0
$263$	31.0969	1.91752	0.958760	−	0.284218i	$-0.0917341\pi$
$263$	31.0969	1.91752	0.958760	+	0.284218i	$0.0917341\pi$
$264$	0	0
$265$	−9.26228	−0.568977
$266$	0	0
$267$	−13.8518	−0.847714
$268$	0	0
$269$	−1.23733	−0.0754411	−0.0377205	−	0.999288i	$-0.512010\pi$
$269$	−1.23733	−0.0754411	−0.0377205	+	0.999288i	$0.512010\pi$
$270$	0	0
$271$	−9.24471	−0.561576	−0.280788	−	0.959770i	$-0.590596\pi$
$271$	−9.24471	−0.561576	−0.280788	+	0.959770i	$0.590596\pi$
$272$	0	0
$273$	−18.8572	−1.14129
$274$	0	0
$275$	7.52989	0.454069
$276$	0	0
$277$	6.13491	0.368611	0.184305	−	0.982869i	$-0.440996\pi$
$277$	6.13491	0.368611	0.184305	+	0.982869i	$0.440996\pi$
$278$	0	0
$279$	10.8446	0.649248
$280$	0	0
$281$	9.93658	0.592766	0.296383	−	0.955069i	$-0.404219\pi$
$281$	9.93658	0.592766	0.296383	+	0.955069i	$0.404219\pi$
$282$	0	0
$283$	4.55893	0.271000	0.135500	−	0.990777i	$-0.456736\pi$
$283$	4.55893	0.271000	0.135500	+	0.990777i	$0.456736\pi$
$284$	0	0
$285$	−5.24773	−0.310849
$286$	0	0
$287$	−10.9188	−0.644514
$288$	0	0
$289$	2.74796	0.161645
$290$	0	0
$291$	−14.0653	−0.824521
$292$	0	0
$293$	−20.4630	−1.19546	−0.597731	−	0.801696i	$-0.703931\pi$
$293$	−20.4630	−1.19546	−0.597731	+	0.801696i	$0.703931\pi$
$294$	0	0
$295$	0.564074	0.0328417
$296$	0	0
$297$	−2.04384	−0.118595
$298$	0	0
$299$	−47.2517	−2.73263
$300$	0	0
$301$	−33.6336	−1.93861
$302$	0	0
$303$	9.38072	0.538908
$304$	0	0
$305$	2.21883	0.127050
$306$	0	0
$307$	9.39972	0.536470	0.268235	−	0.963353i	$-0.413560\pi$
$307$	9.39972	0.536470	0.268235	+	0.963353i	$0.413560\pi$
$308$	0	0
$309$	13.9780	0.795178
$310$	0	0
$311$	9.83249	0.557549	0.278775	−	0.960357i	$-0.410072\pi$
$311$	9.83249	0.557549	0.278775	+	0.960357i	$0.410072\pi$
$312$	0	0
$313$	17.3580	0.981133	0.490567	−	0.871404i	$-0.336790\pi$
$313$	17.3580	0.981133	0.490567	+	0.871404i	$0.336790\pi$
$314$	0	0
$315$	3.81340	0.214861
$316$	0	0
$317$	−6.84251	−0.384314	−0.192157	−	0.981364i	$-0.561548\pi$
$317$	−6.84251	−0.384314	−0.192157	+	0.981364i	$0.561548\pi$
$318$	0	0
$319$	12.4591	0.697573
$320$	0	0
$321$	−8.32030	−0.464394
$322$	0	0
$323$	−20.3299	−1.13119
$324$	0	0
$325$	−20.8980	−1.15921
$326$	0	0
$327$	16.2370	0.897906
$328$	0	0
$329$	−24.0155	−1.32402
$330$	0	0
$331$	−8.53876	−0.469333	−0.234666	−	0.972076i	$-0.575400\pi$
$331$	−8.53876	−0.469333	−0.234666	+	0.972076i	$0.575400\pi$
$332$	0	0
$333$	3.37692	0.185054
$334$	0	0
$335$	−6.61791	−0.361575
$336$	0	0
$337$	13.5786	0.739675	0.369838	−	0.929096i	$-0.379413\pi$
$337$	13.5786	0.739675	0.369838	+	0.929096i	$0.379413\pi$
$338$	0	0
$339$	10.0624	0.546517
$340$	0	0
$341$	−22.1645	−1.20028
$342$	0	0
$343$	9.80112	0.529211
$344$	0	0
$345$	9.55547	0.514449
$346$	0	0
$347$	−29.8672	−1.60335	−0.801677	−	0.597757i	$-0.796059\pi$
$347$	−29.8672	−1.60335	−0.801677	+	0.597757i	$0.796059\pi$
$348$	0	0
$349$	19.2769	1.03187	0.515935	−	0.856627i	$-0.327444\pi$
$349$	19.2769	1.03187	0.515935	+	0.856627i	$0.327444\pi$
$350$	0	0
$351$	5.67233	0.302766
$352$	0	0
$353$	21.4419	1.14124	0.570618	−	0.821215i	$-0.306704\pi$
$353$	21.4419	1.14124	0.570618	+	0.821215i	$0.306704\pi$
$354$	0	0
$355$	−9.66358	−0.512890
$356$	0	0
$357$	14.7733	0.781886
$358$	0	0
$359$	33.2799	1.75645	0.878223	−	0.478251i	$-0.158729\pi$
$359$	33.2799	1.75645	0.878223	+	0.478251i	$0.158729\pi$
$360$	0	0
$361$	1.92909	0.101531
$362$	0	0
$363$	−6.82273	−0.358100
$364$	0	0
$365$	16.8094	0.879844
$366$	0	0
$367$	13.9046	0.725814	0.362907	−	0.931825i	$-0.381784\pi$
$367$	13.9046	0.725814	0.362907	+	0.931825i	$0.381784\pi$
$368$	0	0
$369$	3.28441	0.170980
$370$	0	0
$371$	−26.8434	−1.39364
$372$	0	0
$373$	33.5870	1.73907	0.869535	−	0.493871i	$-0.164418\pi$
$373$	33.5870	1.73907	0.869535	+	0.493871i	$0.164418\pi$
$374$	0	0
$375$	9.96152	0.514411
$376$	0	0
$377$	−34.5780	−1.78086
$378$	0	0
$379$	−5.30400	−0.272448	−0.136224	−	0.990678i	$-0.543497\pi$
$379$	−5.30400	−0.272448	−0.136224	+	0.990678i	$0.543497\pi$
$380$	0	0
$381$	−6.58874	−0.337551
$382$	0	0
$383$	15.4562	0.789777	0.394889	−	0.918729i	$-0.370783\pi$
$383$	15.4562	0.789777	0.394889	+	0.918729i	$0.370783\pi$
$384$	0	0
$385$	−7.79397	−0.397217
$386$	0	0
$387$	10.1171	0.514282
$388$	0	0
$389$	−30.0960	−1.52593	−0.762964	−	0.646441i	$-0.776256\pi$
$389$	−30.0960	−1.52593	−0.762964	+	0.646441i	$0.776256\pi$
$390$	0	0
$391$	37.0183	1.87210
$392$	0	0
$393$	−3.88569	−0.196007
$394$	0	0
$395$	−0.734252	−0.0369442
$396$	0	0
$397$	−32.4774	−1.62999	−0.814997	−	0.579465i	$-0.803262\pi$
$397$	−32.4774	−1.62999	−0.814997	+	0.579465i	$0.803262\pi$
$398$	0	0
$399$	−15.2087	−0.761386
$400$	0	0
$401$	4.25313	0.212391	0.106196	−	0.994345i	$-0.466133\pi$
$401$	4.25313	0.212391	0.106196	+	0.994345i	$0.466133\pi$
$402$	0	0
$403$	61.5140	3.06423
$404$	0	0
$405$	−1.14709	−0.0569992
$406$	0	0
$407$	−6.90187	−0.342113
$408$	0	0
$409$	−27.5145	−1.36050	−0.680252	−	0.732979i	$-0.738129\pi$
$409$	−27.5145	−1.36050	−0.680252	+	0.732979i	$0.738129\pi$
$410$	0	0
$411$	20.3679	1.00468
$412$	0	0
$413$	1.63477	0.0804417
$414$	0	0
$415$	−14.8996	−0.731395
$416$	0	0
$417$	14.2985	0.700199
$418$	0	0
$419$	2.53824	0.124001	0.0620005	−	0.998076i	$-0.480252\pi$
$419$	2.53824	0.124001	0.0620005	+	0.998076i	$0.480252\pi$
$420$	0	0
$421$	−28.9312	−1.41002	−0.705009	−	0.709198i	$-0.749057\pi$
$421$	−28.9312	−1.41002	−0.705009	+	0.709198i	$0.749057\pi$
$422$	0	0
$423$	7.22397	0.351242
$424$	0	0
$425$	16.3721	0.794162
$426$	0	0
$427$	6.43050	0.311194
$428$	0	0
$429$	−11.5933	−0.559731
$430$	0	0
$431$	18.5039	0.891301	0.445651	−	0.895207i	$-0.352972\pi$
$431$	18.5039	0.891301	0.445651	+	0.895207i	$0.352972\pi$
$432$	0	0
$433$	−19.0976	−0.917773	−0.458886	−	0.888495i	$-0.651752\pi$
$433$	−19.0976	−0.917773	−0.458886	+	0.888495i	$0.651752\pi$
$434$	0	0
$435$	6.99254	0.335267
$436$	0	0
$437$	−38.1093	−1.82301
$438$	0	0
$439$	−38.8543	−1.85441	−0.927207	−	0.374548i	$-0.877798\pi$
$439$	−38.8543	−1.85441	−0.927207	+	0.374548i	$0.877798\pi$
$440$	0	0
$441$	4.05178	0.192942
$442$	0	0
$443$	11.4378	0.543426	0.271713	−	0.962378i	$-0.412410\pi$
$443$	11.4378	0.543426	0.271713	+	0.962378i	$0.412410\pi$
$444$	0	0
$445$	15.8892	0.753219
$446$	0	0
$447$	13.5233	0.639631
$448$	0	0
$449$	−13.7933	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$449$	−13.7933	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$450$	0	0
$451$	−6.71280	−0.316093
$452$	0	0
$453$	2.39755	0.112647
$454$	0	0
$455$	21.6309	1.01407
$456$	0	0
$457$	−6.78493	−0.317386	−0.158693	−	0.987328i	$-0.550728\pi$
$457$	−6.78493	−0.317386	−0.158693	+	0.987328i	$0.550728\pi$
$458$	0	0
$459$	−4.44387	−0.207422
$460$	0	0
$461$	−34.5307	−1.60826	−0.804128	−	0.594457i	$-0.797367\pi$
$461$	−34.5307	−1.60826	−0.804128	+	0.594457i	$0.797367\pi$
$462$	0	0
$463$	3.97269	0.184627	0.0923133	−	0.995730i	$-0.470574\pi$
$463$	3.97269	0.184627	0.0923133	+	0.995730i	$0.470574\pi$
$464$	0	0
$465$	−12.4397	−0.576876
$466$	0	0
$467$	−26.7295	−1.23690	−0.618448	−	0.785826i	$-0.712238\pi$
$467$	−26.7295	−1.23690	−0.618448	+	0.785826i	$0.712238\pi$
$468$	0	0
$469$	−19.1797	−0.885635
$470$	0	0
$471$	−1.40129	−0.0645679
$472$	0	0
$473$	−20.6778	−0.950764
$474$	0	0
$475$	−16.8546	−0.773340
$476$	0	0
$477$	8.07461	0.369711
$478$	0	0
$479$	−6.79972	−0.310687	−0.155343	−	0.987861i	$-0.549648\pi$
$479$	−6.79972	−0.310687	−0.155343	+	0.987861i	$0.549648\pi$
$480$	0	0
$481$	19.1550	0.873392
$482$	0	0
$483$	27.6931	1.26008
$484$	0	0
$485$	16.1341	0.732611
$486$	0	0
$487$	−7.75129	−0.351244	−0.175622	−	0.984458i	$-0.556194\pi$
$487$	−7.75129	−0.351244	−0.175622	+	0.984458i	$0.556194\pi$
$488$	0	0
$489$	13.9978	0.633003
$490$	0	0
$491$	25.4942	1.15054	0.575269	−	0.817964i	$-0.304897\pi$
$491$	25.4942	1.15054	0.575269	+	0.817964i	$0.304897\pi$
$492$	0	0
$493$	27.0894	1.22005
$494$	0	0
$495$	2.34446	0.105376
$496$	0	0
$497$	−28.0065	−1.25626
$498$	0	0
$499$	10.0719	0.450879	0.225440	−	0.974257i	$-0.427618\pi$
$499$	10.0719	0.450879	0.225440	+	0.974257i	$0.427618\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	27.6005	1.23065	0.615323	−	0.788275i	$-0.289026\pi$
$503$	27.6005	1.23065	0.615323	+	0.788275i	$0.289026\pi$
$504$	0	0
$505$	−10.7605	−0.478836
$506$	0	0
$507$	19.1753	0.851605
$508$	0	0
$509$	6.46064	0.286363	0.143181	−	0.989696i	$-0.454267\pi$
$509$	6.46064	0.286363	0.143181	+	0.989696i	$0.454267\pi$
$510$	0	0
$511$	48.7161	2.15507
$512$	0	0
$513$	4.57483	0.201984
$514$	0	0
$515$	−16.0339	−0.706539
$516$	0	0
$517$	−14.7646	−0.649348
$518$	0	0
$519$	2.70111	0.118565
$520$	0	0
$521$	−1.33945	−0.0586822	−0.0293411	−	0.999569i	$-0.509341\pi$
$521$	−1.33945	−0.0586822	−0.0293411	+	0.999569i	$0.509341\pi$
$522$	0	0
$523$	−32.4828	−1.42037	−0.710187	−	0.704013i	$-0.751390\pi$
$523$	−32.4828	−1.42037	−0.710187	+	0.704013i	$0.751390\pi$
$524$	0	0
$525$	12.2478	0.534538
$526$	0	0
$527$	−48.1918	−2.09927
$528$	0	0
$529$	46.3923	2.01706
$530$	0	0
$531$	−0.491745	−0.0213399
$532$	0	0
$533$	18.6302	0.806966
$534$	0	0
$535$	9.54410	0.412627
$536$	0	0
$537$	−2.99486	−0.129238
$538$	0	0
$539$	−8.28119	−0.356696
$540$	0	0
$541$	6.51171	0.279960	0.139980	−	0.990154i	$-0.455296\pi$
$541$	6.51171	0.279960	0.139980	+	0.990154i	$0.455296\pi$
$542$	0	0
$543$	13.5390	0.581012
$544$	0	0
$545$	−18.6252	−0.797816
$546$	0	0
$547$	16.4574	0.703669	0.351835	−	0.936062i	$-0.385558\pi$
$547$	16.4574	0.703669	0.351835	+	0.936062i	$0.385558\pi$
$548$	0	0
$549$	−1.93432	−0.0825548
$550$	0	0
$551$	−27.8878	−1.18806
$552$	0	0
$553$	−2.12797	−0.0904905
$554$	0	0
$555$	−3.87362	−0.164426
$556$	0	0
$557$	23.1395	0.980453	0.490226	−	0.871595i	$-0.336914\pi$
$557$	23.1395	0.980453	0.490226	+	0.871595i	$0.336914\pi$
$558$	0	0
$559$	57.3877	2.42724
$560$	0	0
$561$	9.08254	0.383465
$562$	0	0
$563$	−18.7483	−0.790146	−0.395073	−	0.918650i	$-0.629281\pi$
$563$	−18.7483	−0.790146	−0.395073	+	0.918650i	$0.629281\pi$
$564$	0	0
$565$	−11.5425	−0.485596
$566$	0	0
$567$	−3.32442	−0.139613
$568$	0	0
$569$	−30.0098	−1.25808	−0.629038	−	0.777375i	$-0.716551\pi$
$569$	−30.0098	−1.25808	−0.629038	+	0.777375i	$0.716551\pi$
$570$	0	0
$571$	−26.4171	−1.10552	−0.552760	−	0.833341i	$-0.686425\pi$
$571$	−26.4171	−1.10552	−0.552760	+	0.833341i	$0.686425\pi$
$572$	0	0
$573$	−18.3703	−0.767429
$574$	0	0
$575$	30.6901	1.27986
$576$	0	0
$577$	−4.92377	−0.204979	−0.102490	−	0.994734i	$-0.532681\pi$
$577$	−4.92377	−0.204979	−0.102490	+	0.994734i	$0.532681\pi$
$578$	0	0
$579$	−8.15030	−0.338715
$580$	0	0
$581$	−43.1813	−1.79146
$582$	0	0
$583$	−16.5032	−0.683493
$584$	0	0
$585$	−6.50665	−0.269017
$586$	0	0
$587$	−10.1452	−0.418738	−0.209369	−	0.977837i	$-0.567141\pi$
$587$	−10.1452	−0.418738	−0.209369	+	0.977837i	$0.567141\pi$
$588$	0	0
$589$	49.6121	2.04423
$590$	0	0
$591$	9.56487	0.393446
$592$	0	0
$593$	2.97336	0.122101	0.0610507	−	0.998135i	$-0.480555\pi$
$593$	2.97336	0.122101	0.0610507	+	0.998135i	$0.480555\pi$
$594$	0	0
$595$	−16.9462	−0.694728
$596$	0	0
$597$	−24.9138	−1.01965
$598$	0	0
$599$	−1.29605	−0.0529553	−0.0264777	−	0.999649i	$-0.508429\pi$
$599$	−1.29605	−0.0529553	−0.0264777	+	0.999649i	$0.508429\pi$
$600$	0	0
$601$	−19.8043	−0.807836	−0.403918	−	0.914795i	$-0.632352\pi$
$601$	−19.8043	−0.807836	−0.403918	+	0.914795i	$0.632352\pi$
$602$	0	0
$603$	5.76932	0.234945
$604$	0	0
$605$	7.82626	0.318183
$606$	0	0
$607$	−34.9075	−1.41685	−0.708425	−	0.705786i	$-0.750594\pi$
$607$	−34.9075	−1.41685	−0.708425	+	0.705786i	$0.750594\pi$
$608$	0	0
$609$	20.2654	0.821195
$610$	0	0
$611$	40.9767	1.65774
$612$	0	0
$613$	8.54783	0.345244	0.172622	−	0.984988i	$-0.444776\pi$
$613$	8.54783	0.345244	0.172622	+	0.984988i	$0.444776\pi$
$614$	0	0
$615$	−3.76750	−0.151920
$616$	0	0
$617$	−31.5796	−1.27135	−0.635673	−	0.771958i	$-0.719277\pi$
$617$	−31.5796	−1.27135	−0.635673	+	0.771958i	$0.719277\pi$
$618$	0	0
$619$	−8.99449	−0.361519	−0.180760	−	0.983527i	$-0.557856\pi$
$619$	−8.99449	−0.361519	−0.180760	+	0.983527i	$0.557856\pi$
$620$	0	0
$621$	−8.33021	−0.334280
$622$	0	0
$623$	46.0491	1.84492
$624$	0	0
$625$	6.99424	0.279770
$626$	0	0
$627$	−9.35021	−0.373412
$628$	0	0
$629$	−15.0066	−0.598352
$630$	0	0
$631$	−27.2504	−1.08482	−0.542411	−	0.840114i	$-0.682488\pi$
$631$	−27.2504	−1.08482	−0.542411	+	0.840114i	$0.682488\pi$
$632$	0	0
$633$	21.2970	0.846480
$634$	0	0
$635$	7.55786	0.299924
$636$	0	0
$637$	22.9830	0.910621
$638$	0	0
$639$	8.42446	0.333266
$640$	0	0
$641$	40.2440	1.58954	0.794772	−	0.606909i	$-0.207591\pi$
$641$	40.2440	1.58954	0.794772	+	0.606909i	$0.207591\pi$
$642$	0	0
$643$	7.20369	0.284086	0.142043	−	0.989861i	$-0.454633\pi$
$643$	7.20369	0.284086	0.142043	+	0.989861i	$0.454633\pi$
$644$	0	0
$645$	−11.6052	−0.456955
$646$	0	0
$647$	−19.8491	−0.780349	−0.390175	−	0.920741i	$-0.627585\pi$
$647$	−19.8491	−0.780349	−0.390175	+	0.920741i	$0.627585\pi$
$648$	0	0
$649$	1.00505	0.0394516
$650$	0	0
$651$	−36.0519	−1.41299
$652$	0	0
$653$	27.0947	1.06030	0.530149	−	0.847905i	$-0.322136\pi$
$653$	27.0947	1.06030	0.530149	+	0.847905i	$0.322136\pi$
$654$	0	0
$655$	4.45722	0.174158
$656$	0	0
$657$	−14.6540	−0.571707
$658$	0	0
$659$	−0.316737	−0.0123383	−0.00616916	−	0.999981i	$-0.501964\pi$
$659$	−0.316737	−0.0123383	−0.00616916	+	0.999981i	$0.501964\pi$
$660$	0	0
$661$	4.11685	0.160127	0.0800635	−	0.996790i	$-0.474488\pi$
$661$	4.11685	0.160127	0.0800635	+	0.996790i	$0.474488\pi$
$662$	0	0
$663$	−25.2071	−0.978962
$664$	0	0
$665$	17.4457	0.676514
$666$	0	0
$667$	50.7802	1.96622
$668$	0	0
$669$	21.5679	0.833865
$670$	0	0
$671$	3.95344	0.152621
$672$	0	0
$673$	−18.6970	−0.720718	−0.360359	−	0.932814i	$-0.617346\pi$
$673$	−18.6970	−0.720718	−0.360359	+	0.932814i	$0.617346\pi$
$674$	0	0
$675$	−3.68419	−0.141805
$676$	0	0
$677$	7.14151	0.274470	0.137235	−	0.990538i	$-0.456178\pi$
$677$	7.14151	0.274470	0.137235	+	0.990538i	$0.456178\pi$
$678$	0	0
$679$	46.7589	1.79444
$680$	0	0
$681$	23.9038	0.915995
$682$	0	0
$683$	−17.6229	−0.674323	−0.337162	−	0.941447i	$-0.609467\pi$
$683$	−17.6229	−0.674323	−0.337162	+	0.941447i	$0.609467\pi$
$684$	0	0
$685$	−23.3638	−0.892685
$686$	0	0
$687$	5.37332	0.205005
$688$	0	0
$689$	45.8019	1.74491
$690$	0	0
$691$	−12.2885	−0.467476	−0.233738	−	0.972300i	$-0.575096\pi$
$691$	−12.2885	−0.467476	−0.233738	+	0.972300i	$0.575096\pi$
$692$	0	0
$693$	6.79458	0.258105
$694$	0	0
$695$	−16.4016	−0.622147
$696$	0	0
$697$	−14.5955	−0.552843
$698$	0	0
$699$	−22.0472	−0.833900
$700$	0	0
$701$	3.27369	0.123646	0.0618229	−	0.998087i	$-0.480309\pi$
$701$	3.27369	0.123646	0.0618229	+	0.998087i	$0.480309\pi$
$702$	0	0
$703$	15.4488	0.582664
$704$	0	0
$705$	−8.28652	−0.312088
$706$	0	0
$707$	−31.1855	−1.17285
$708$	0	0
$709$	−33.5377	−1.25953	−0.629767	−	0.776784i	$-0.716850\pi$
$709$	−33.5377	−1.25953	−0.629767	+	0.776784i	$0.716850\pi$
$710$	0	0
$711$	0.640102	0.0240057
$712$	0	0
$713$	−90.3375	−3.38317
$714$	0	0
$715$	13.2985	0.497337
$716$	0	0
$717$	−11.1537	−0.416543
$718$	0	0
$719$	42.1058	1.57028	0.785141	−	0.619317i	$-0.212590\pi$
$719$	42.1058	1.57028	0.785141	+	0.619317i	$0.212590\pi$
$720$	0	0
$721$	−46.4686	−1.73058
$722$	0	0
$723$	−6.82085	−0.253670
$724$	0	0
$725$	22.4585	0.834088
$726$	0	0
$727$	−31.1152	−1.15400	−0.576999	−	0.816745i	$-0.695776\pi$
$727$	−31.1152	−1.15400	−0.576999	+	0.816745i	$0.695776\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−44.9592	−1.66287
$732$	0	0
$733$	17.6036	0.650202	0.325101	−	0.945679i	$-0.394602\pi$
$733$	17.6036	0.650202	0.325101	+	0.945679i	$0.394602\pi$
$734$	0	0
$735$	−4.64775	−0.171435
$736$	0	0
$737$	−11.7916	−0.434348
$738$	0	0
$739$	31.9083	1.17377	0.586883	−	0.809672i	$-0.300355\pi$
$739$	31.9083	1.17377	0.586883	+	0.809672i	$0.300355\pi$
$740$	0	0
$741$	25.9500	0.953295
$742$	0	0
$743$	43.2538	1.58683	0.793415	−	0.608681i	$-0.208301\pi$
$743$	43.2538	1.58683	0.793415	+	0.608681i	$0.208301\pi$
$744$	0	0
$745$	−15.5124	−0.568331
$746$	0	0
$747$	12.9891	0.475247
$748$	0	0
$749$	27.6602	1.01068
$750$	0	0
$751$	24.4441	0.891978	0.445989	−	0.895038i	$-0.352852\pi$
$751$	24.4441	0.891978	0.445989	+	0.895038i	$0.352852\pi$
$752$	0	0
$753$	−14.3413	−0.522627
$754$	0	0
$755$	−2.75020	−0.100090
$756$	0	0
$757$	−6.19894	−0.225304	−0.112652	−	0.993634i	$-0.535935\pi$
$757$	−6.19894	−0.225304	−0.112652	+	0.993634i	$0.535935\pi$
$758$	0	0
$759$	17.0256	0.617990
$760$	0	0
$761$	9.28311	0.336512	0.168256	−	0.985743i	$-0.446186\pi$
$761$	9.28311	0.336512	0.168256	+	0.985743i	$0.446186\pi$
$762$	0	0
$763$	−53.9785	−1.95415
$764$	0	0
$765$	5.09750	0.184301
$766$	0	0
$767$	−2.78934	−0.100717
$768$	0	0
$769$	0.140927	0.00508195	0.00254098	−	0.999997i	$-0.499191\pi$
$769$	0.140927	0.00508195	0.00254098	+	0.999997i	$0.499191\pi$
$770$	0	0
$771$	−6.18565	−0.222771
$772$	0	0
$773$	18.1051	0.651193	0.325597	−	0.945509i	$-0.394435\pi$
$773$	18.1051	0.651193	0.325597	+	0.945509i	$0.394435\pi$
$774$	0	0
$775$	−39.9535	−1.43517
$776$	0	0
$777$	−11.2263	−0.402741
$778$	0	0
$779$	15.0256	0.538349
$780$	0	0
$781$	−17.2182	−0.616117
$782$	0	0
$783$	−6.09591	−0.217850
$784$	0	0
$785$	1.60740	0.0573705
$786$	0	0
$787$	−18.1360	−0.646478	−0.323239	−	0.946317i	$-0.604772\pi$
$787$	−18.1360	−0.646478	−0.323239	+	0.946317i	$0.604772\pi$
$788$	0	0
$789$	31.0969	1.10708
$790$	0	0
$791$	−33.4518	−1.18941
$792$	0	0
$793$	−10.9721	−0.389631
$794$	0	0
$795$	−9.26228	−0.328499
$796$	0	0
$797$	3.43944	0.121831	0.0609157	−	0.998143i	$-0.480598\pi$
$797$	3.43944	0.121831	0.0609157	+	0.998143i	$0.480598\pi$
$798$	0	0
$799$	−32.1024	−1.13570
$800$	0	0
$801$	−13.8518	−0.489428
$802$	0	0
$803$	29.9504	1.05693
$804$	0	0
$805$	−31.7664	−1.11962
$806$	0	0
$807$	−1.23733	−0.0435559
$808$	0	0
$809$	−34.3884	−1.20903	−0.604516	−	0.796593i	$-0.706633\pi$
$809$	−34.3884	−1.20903	−0.604516	+	0.796593i	$0.706633\pi$
$810$	0	0
$811$	−30.4085	−1.06779	−0.533894	−	0.845552i	$-0.679272\pi$
$811$	−30.4085	−1.06779	−0.533894	+	0.845552i	$0.679272\pi$
$812$	0	0
$813$	−9.24471	−0.324226
$814$	0	0
$815$	−16.0567	−0.562442
$816$	0	0
$817$	46.2841	1.61928
$818$	0	0
$819$	−18.8572	−0.658924
$820$	0	0
$821$	8.43876	0.294515	0.147257	−	0.989098i	$-0.452955\pi$
$821$	8.43876	0.294515	0.147257	+	0.989098i	$0.452955\pi$
$822$	0	0
$823$	−49.0541	−1.70992	−0.854960	−	0.518694i	$-0.826418\pi$
$823$	−49.0541	−1.70992	−0.854960	+	0.518694i	$0.826418\pi$
$824$	0	0
$825$	7.52989	0.262157
$826$	0	0
$827$	−52.8282	−1.83702	−0.918508	−	0.395401i	$-0.870606\pi$
$827$	−52.8282	−1.83702	−0.918508	+	0.395401i	$0.870606\pi$
$828$	0	0
$829$	45.6094	1.58408	0.792041	−	0.610468i	$-0.209019\pi$
$829$	45.6094	1.58408	0.792041	+	0.610468i	$0.209019\pi$
$830$	0	0
$831$	6.13491	0.212818
$832$	0	0
$833$	−18.0056	−0.623857
$834$	0	0
$835$	−1.14709	−0.0396966
$836$	0	0
$837$	10.8446	0.374843
$838$	0	0
$839$	7.63655	0.263643	0.131821	−	0.991273i	$-0.457917\pi$
$839$	7.63655	0.263643	0.131821	+	0.991273i	$0.457917\pi$
$840$	0	0
$841$	8.16013	0.281384
$842$	0	0
$843$	9.93658	0.342234
$844$	0	0
$845$	−21.9957	−0.756676
$846$	0	0
$847$	22.6816	0.779350
$848$	0	0
$849$	4.55893	0.156462
$850$	0	0
$851$	−28.1304	−0.964299
$852$	0	0
$853$	−41.6725	−1.42684	−0.713420	−	0.700737i	$-0.752855\pi$
$853$	−41.6725	−1.42684	−0.713420	+	0.700737i	$0.752855\pi$
$854$	0	0
$855$	−5.24773	−0.179469
$856$	0	0
$857$	35.6503	1.21779	0.608895	−	0.793251i	$-0.291613\pi$
$857$	35.6503	1.21779	0.608895	+	0.793251i	$0.291613\pi$
$858$	0	0
$859$	25.6619	0.875574	0.437787	−	0.899079i	$-0.355762\pi$
$859$	25.6619	0.875574	0.437787	+	0.899079i	$0.355762\pi$
$860$	0	0
$861$	−10.9188	−0.372110
$862$	0	0
$863$	37.4463	1.27469	0.637344	−	0.770580i	$-0.280033\pi$
$863$	37.4463	1.27469	0.637344	+	0.770580i	$0.280033\pi$
$864$	0	0
$865$	−3.09840	−0.105349
$866$	0	0
$867$	2.74796	0.0933257
$868$	0	0
$869$	−1.30826	−0.0443798
$870$	0	0
$871$	32.7255	1.10886
$872$	0	0
$873$	−14.0653	−0.476037
$874$	0	0
$875$	−33.1163	−1.11954
$876$	0	0
$877$	−29.5206	−0.996839	−0.498420	−	0.866936i	$-0.666086\pi$
$877$	−29.5206	−0.996839	−0.498420	+	0.866936i	$0.666086\pi$
$878$	0	0
$879$	−20.4630	−0.690201
$880$	0	0
$881$	24.7820	0.834927	0.417464	−	0.908694i	$-0.362919\pi$
$881$	24.7820	0.834927	0.417464	+	0.908694i	$0.362919\pi$
$882$	0	0
$883$	51.0962	1.71952	0.859762	−	0.510695i	$-0.170612\pi$
$883$	51.0962	1.71952	0.859762	+	0.510695i	$0.170612\pi$
$884$	0	0
$885$	0.564074	0.0189611
$886$	0	0
$887$	−1.22606	−0.0411671	−0.0205836	−	0.999788i	$-0.506552\pi$
$887$	−1.22606	−0.0411671	−0.0205836	+	0.999788i	$0.506552\pi$
$888$	0	0
$889$	21.9038	0.734628
$890$	0	0
$891$	−2.04384	−0.0684711
$892$	0	0
$893$	33.0485	1.10592
$894$	0	0
$895$	3.43536	0.114832
$896$	0	0
$897$	−47.2517	−1.57769
$898$	0	0
$899$	−66.1076	−2.20481
$900$	0	0
$901$	−35.8825	−1.19542
$902$	0	0
$903$	−33.6336	−1.11926
$904$	0	0
$905$	−15.5304	−0.516246
$906$	0	0
$907$	36.2593	1.20397	0.601985	−	0.798508i	$-0.294377\pi$
$907$	36.2593	1.20397	0.601985	+	0.798508i	$0.294377\pi$
$908$	0	0
$909$	9.38072	0.311139
$910$	0	0
$911$	2.31720	0.0767724	0.0383862	−	0.999263i	$-0.487778\pi$
$911$	2.31720	0.0767724	0.0383862	+	0.999263i	$0.487778\pi$
$912$	0	0
$913$	−26.5477	−0.878599
$914$	0	0
$915$	2.21883	0.0733524
$916$	0	0
$917$	12.9177	0.426579
$918$	0	0
$919$	−26.6598	−0.879427	−0.439714	−	0.898138i	$-0.644920\pi$
$919$	−26.6598	−0.879427	−0.439714	+	0.898138i	$0.644920\pi$
$920$	0	0
$921$	9.39972	0.309731
$922$	0	0
$923$	47.7863	1.57291
$924$	0	0
$925$	−12.4412	−0.409065
$926$	0	0
$927$	13.9780	0.459096
$928$	0	0
$929$	−14.5860	−0.478553	−0.239276	−	0.970952i	$-0.576910\pi$
$929$	−14.5860	−0.478553	−0.239276	+	0.970952i	$0.576910\pi$
$930$	0	0
$931$	18.5362	0.607500
$932$	0	0
$933$	9.83249	0.321901
$934$	0	0
$935$	−10.4185	−0.340720
$936$	0	0
$937$	52.2258	1.70614	0.853071	−	0.521794i	$-0.174737\pi$
$937$	52.2258	1.70614	0.853071	+	0.521794i	$0.174737\pi$
$938$	0	0
$939$	17.3580	0.566458
$940$	0	0
$941$	35.7136	1.16423	0.582115	−	0.813106i	$-0.302225\pi$
$941$	35.7136	1.16423	0.582115	+	0.813106i	$0.302225\pi$
$942$	0	0
$943$	−27.3598	−0.890958
$944$	0	0
$945$	3.81340	0.124050
$946$	0	0
$947$	25.9066	0.841851	0.420926	−	0.907095i	$-0.361705\pi$
$947$	25.9066	0.841851	0.420926	+	0.907095i	$0.361705\pi$
$948$	0	0
$949$	−83.1222	−2.69826
$950$	0	0
$951$	−6.84251	−0.221884
$952$	0	0
$953$	37.1660	1.20393	0.601963	−	0.798524i	$-0.294386\pi$
$953$	37.1660	1.20393	0.601963	+	0.798524i	$0.294386\pi$
$954$	0	0
$955$	21.0723	0.681883
$956$	0	0
$957$	12.4591	0.402744
$958$	0	0
$959$	−67.7116	−2.18652
$960$	0	0
$961$	86.6048	2.79370
$962$	0	0
$963$	−8.32030	−0.268118
$964$	0	0
$965$	9.34910	0.300958
$966$	0	0
$967$	−2.00936	−0.0646165	−0.0323083	−	0.999478i	$-0.510286\pi$
$967$	−2.00936	−0.0646165	−0.0323083	+	0.999478i	$0.510286\pi$
$968$	0	0
$969$	−20.3299	−0.653092
$970$	0	0
$971$	−60.4891	−1.94119	−0.970594	−	0.240722i	$-0.922616\pi$
$971$	−60.4891	−1.94119	−0.970594	+	0.240722i	$0.922616\pi$
$972$	0	0
$973$	−47.5341	−1.52387
$974$	0	0
$975$	−20.8980	−0.669270
$976$	0	0
$977$	−12.8429	−0.410880	−0.205440	−	0.978670i	$-0.565863\pi$
$977$	−12.8429	−0.410880	−0.205440	+	0.978670i	$0.565863\pi$
$978$	0	0
$979$	28.3107	0.904815
$980$	0	0
$981$	16.2370	0.518406
$982$	0	0
$983$	53.1897	1.69649	0.848243	−	0.529607i	$-0.177661\pi$
$983$	53.1897	1.69649	0.848243	+	0.529607i	$0.177661\pi$
$984$	0	0
$985$	−10.9717	−0.349589
$986$	0	0
$987$	−24.0155	−0.764423
$988$	0	0
$989$	−84.2777	−2.67988
$990$	0	0
$991$	21.7567	0.691124	0.345562	−	0.938396i	$-0.387688\pi$
$991$	21.7567	0.691124	0.345562	+	0.938396i	$0.387688\pi$
$992$	0	0
$993$	−8.53876	−0.270969
$994$	0	0
$995$	28.5782	0.905991
$996$	0	0
$997$	−1.56973	−0.0497137	−0.0248569	−	0.999691i	$-0.507913\pi$
$997$	−1.56973	−0.0497137	−0.0248569	+	0.999691i	$0.507913\pi$
$998$	0	0
$999$	3.37692	0.106841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.4	✓	12
4.3	odd	2		8016.2.a.bf.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.4	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.4		12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.14709 q^{5} -3.32442 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.14709 q^{5} -3.32442 q^{7} +1.00000 q^{9} -2.04384 q^{11} +5.67233 q^{13} -1.14709 q^{15} -4.44387 q^{17} +4.57483 q^{19} -3.32442 q^{21} -8.33021 q^{23} -3.68419 q^{25} +1.00000 q^{27} -6.09591 q^{29} +10.8446 q^{31} -2.04384 q^{33} +3.81340 q^{35} +3.37692 q^{37} +5.67233 q^{39} +3.28441 q^{41} +10.1171 q^{43} -1.14709 q^{45} +7.22397 q^{47} +4.05178 q^{49} -4.44387 q^{51} +8.07461 q^{53} +2.34446 q^{55} +4.57483 q^{57} -0.491745 q^{59} -1.93432 q^{61} -3.32442 q^{63} -6.50665 q^{65} +5.76932 q^{67} -8.33021 q^{69} +8.42446 q^{71} -14.6540 q^{73} -3.68419 q^{75} +6.79458 q^{77} +0.640102 q^{79} +1.00000 q^{81} +12.9891 q^{83} +5.09750 q^{85} -6.09591 q^{87} -13.8518 q^{89} -18.8572 q^{91} +10.8446 q^{93} -5.24773 q^{95} -14.0653 q^{97} -2.04384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + q^99

Introduction

L-functions

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Database

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