LMFDB - Embedded newform 4008.2.a.l.1.7

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			1.7
Root			$1.53322$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.53322 q^{5} -0.915565 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.53322 q^{5} -0.915565 q^{7} +1.00000 q^{9} +3.09767 q^{11} +1.39795 q^{13} +1.53322 q^{15} -1.50046 q^{17} -4.21086 q^{19} -0.915565 q^{21} +4.95660 q^{23} -2.64923 q^{25} +1.00000 q^{27} +1.76793 q^{29} +3.79993 q^{31} +3.09767 q^{33} -1.40377 q^{35} +4.15442 q^{37} +1.39795 q^{39} +8.32859 q^{41} +8.45697 q^{43} +1.53322 q^{45} -6.81577 q^{47} -6.16174 q^{49} -1.50046 q^{51} +1.43276 q^{53} +4.74943 q^{55} -4.21086 q^{57} +12.9348 q^{59} -3.76679 q^{61} -0.915565 q^{63} +2.14337 q^{65} +7.30009 q^{67} +4.95660 q^{69} +0.883952 q^{71} -0.891880 q^{73} -2.64923 q^{75} -2.83612 q^{77} -1.12995 q^{79} +1.00000 q^{81} +4.26402 q^{83} -2.30053 q^{85} +1.76793 q^{87} +17.6794 q^{89} -1.27991 q^{91} +3.79993 q^{93} -6.45620 q^{95} -4.28751 q^{97} +3.09767 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.53322	0.685678	0.342839	−	0.939394i	$-0.388611\pi$
$5$	1.53322	0.685678	0.342839	+	0.939394i	$0.388611\pi$
$6$	0	0
$7$	−0.915565	−0.346051	−0.173025	−	0.984917i	$-0.555354\pi$
$7$	−0.915565	−0.346051	−0.173025	+	0.984917i	$0.555354\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.09767	0.933984	0.466992	−	0.884262i	$-0.345338\pi$
$11$	3.09767	0.933984	0.466992	+	0.884262i	$0.345338\pi$
$12$	0	0
$13$	1.39795	0.387721	0.193861	−	0.981029i	$-0.437899\pi$
$13$	1.39795	0.387721	0.193861	+	0.981029i	$0.437899\pi$
$14$	0	0
$15$	1.53322	0.395877
$16$	0	0
$17$	−1.50046	−0.363914	−0.181957	−	0.983306i	$-0.558243\pi$
$17$	−1.50046	−0.363914	−0.181957	+	0.983306i	$0.558243\pi$
$18$	0	0
$19$	−4.21086	−0.966038	−0.483019	−	0.875610i	$-0.660460\pi$
$19$	−4.21086	−0.966038	−0.483019	+	0.875610i	$0.660460\pi$
$20$	0	0
$21$	−0.915565	−0.199793
$22$	0	0
$23$	4.95660	1.03352	0.516761	−	0.856129i	$-0.327137\pi$
$23$	4.95660	1.03352	0.516761	+	0.856129i	$0.327137\pi$
$24$	0	0
$25$	−2.64923	−0.529845
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.76793	0.328297	0.164148	−	0.986436i	$-0.447512\pi$
$29$	1.76793	0.328297	0.164148	+	0.986436i	$0.447512\pi$
$30$	0	0
$31$	3.79993	0.682487	0.341244	−	0.939975i	$-0.389152\pi$
$31$	3.79993	0.682487	0.341244	+	0.939975i	$0.389152\pi$
$32$	0	0
$33$	3.09767	0.539236
$34$	0	0
$35$	−1.40377	−0.237280
$36$	0	0
$37$	4.15442	0.682982	0.341491	−	0.939885i	$-0.389068\pi$
$37$	4.15442	0.682982	0.341491	+	0.939885i	$0.389068\pi$
$38$	0	0
$39$	1.39795	0.223851
$40$	0	0
$41$	8.32859	1.30071	0.650354	−	0.759631i	$-0.274621\pi$
$41$	8.32859	1.30071	0.650354	+	0.759631i	$0.274621\pi$
$42$	0	0
$43$	8.45697	1.28968	0.644838	−	0.764319i	$-0.276925\pi$
$43$	8.45697	1.28968	0.644838	+	0.764319i	$0.276925\pi$
$44$	0	0
$45$	1.53322	0.228559
$46$	0	0
$47$	−6.81577	−0.994182	−0.497091	−	0.867699i	$-0.665598\pi$
$47$	−6.81577	−0.994182	−0.497091	+	0.867699i	$0.665598\pi$
$48$	0	0
$49$	−6.16174	−0.880249
$50$	0	0
$51$	−1.50046	−0.210106
$52$	0	0
$53$	1.43276	0.196804	0.0984021	−	0.995147i	$-0.468627\pi$
$53$	1.43276	0.196804	0.0984021	+	0.995147i	$0.468627\pi$
$54$	0	0
$55$	4.74943	0.640413
$56$	0	0
$57$	−4.21086	−0.557742
$58$	0	0
$59$	12.9348	1.68397	0.841987	−	0.539499i	$-0.181386\pi$
$59$	12.9348	1.68397	0.841987	+	0.539499i	$0.181386\pi$
$60$	0	0
$61$	−3.76679	−0.482288	−0.241144	−	0.970489i	$-0.577523\pi$
$61$	−3.76679	−0.482288	−0.241144	+	0.970489i	$0.577523\pi$
$62$	0	0
$63$	−0.915565	−0.115350
$64$	0	0
$65$	2.14337	0.265852
$66$	0	0
$67$	7.30009	0.891848	0.445924	−	0.895071i	$-0.352875\pi$
$67$	7.30009	0.891848	0.445924	+	0.895071i	$0.352875\pi$
$68$	0	0
$69$	4.95660	0.596705
$70$	0	0
$71$	0.883952	0.104906	0.0524529	−	0.998623i	$-0.483296\pi$
$71$	0.883952	0.104906	0.0524529	+	0.998623i	$0.483296\pi$
$72$	0	0
$73$	−0.891880	−0.104387	−0.0521933	−	0.998637i	$-0.516621\pi$
$73$	−0.891880	−0.104387	−0.0521933	+	0.998637i	$0.516621\pi$
$74$	0	0
$75$	−2.64923	−0.305906
$76$	0	0
$77$	−2.83612	−0.323206
$78$	0	0
$79$	−1.12995	−0.127129	−0.0635645	−	0.997978i	$-0.520247\pi$
$79$	−1.12995	−0.127129	−0.0635645	+	0.997978i	$0.520247\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.26402	0.468037	0.234018	−	0.972232i	$-0.424812\pi$
$83$	4.26402	0.468037	0.234018	+	0.972232i	$0.424812\pi$
$84$	0	0
$85$	−2.30053	−0.249528
$86$	0	0
$87$	1.76793	0.189542
$88$	0	0
$89$	17.6794	1.87401	0.937007	−	0.349311i	$-0.113584\pi$
$89$	17.6794	1.87401	0.937007	+	0.349311i	$0.113584\pi$
$90$	0	0
$91$	−1.27991	−0.134171
$92$	0	0
$93$	3.79993	0.394034
$94$	0	0
$95$	−6.45620	−0.662392
$96$	0	0
$97$	−4.28751	−0.435331	−0.217666	−	0.976023i	$-0.569844\pi$
$97$	−4.28751	−0.435331	−0.217666	+	0.976023i	$0.569844\pi$
$98$	0	0
$99$	3.09767	0.311328
$100$	0	0
$101$	−15.5844	−1.55071	−0.775355	−	0.631526i	$-0.782429\pi$
$101$	−15.5844	−1.55071	−0.775355	+	0.631526i	$0.782429\pi$
$102$	0	0
$103$	−5.71458	−0.563075	−0.281537	−	0.959550i	$-0.590844\pi$
$103$	−5.71458	−0.563075	−0.281537	+	0.959550i	$0.590844\pi$
$104$	0	0
$105$	−1.40377	−0.136993
$106$	0	0
$107$	−3.17924	−0.307349	−0.153675	−	0.988122i	$-0.549111\pi$
$107$	−3.17924	−0.307349	−0.153675	+	0.988122i	$0.549111\pi$
$108$	0	0
$109$	−12.5475	−1.20184	−0.600919	−	0.799310i	$-0.705198\pi$
$109$	−12.5475	−1.20184	−0.600919	+	0.799310i	$0.705198\pi$
$110$	0	0
$111$	4.15442	0.394320
$112$	0	0
$113$	2.50783	0.235916	0.117958	−	0.993019i	$-0.462365\pi$
$113$	2.50783	0.235916	0.117958	+	0.993019i	$0.462365\pi$
$114$	0	0
$115$	7.59958	0.708664
$116$	0	0
$117$	1.39795	0.129240
$118$	0	0
$119$	1.37376	0.125933
$120$	0	0
$121$	−1.40441	−0.127674
$122$	0	0
$123$	8.32859	0.750964
$124$	0	0
$125$	−11.7280	−1.04898
$126$	0	0
$127$	17.3437	1.53900	0.769501	−	0.638646i	$-0.220505\pi$
$127$	17.3437	1.53900	0.769501	+	0.638646i	$0.220505\pi$
$128$	0	0
$129$	8.45697	0.744595
$130$	0	0
$131$	−16.3850	−1.43156	−0.715782	−	0.698324i	$-0.753929\pi$
$131$	−16.3850	−1.43156	−0.715782	+	0.698324i	$0.753929\pi$
$132$	0	0
$133$	3.85532	0.334298
$134$	0	0
$135$	1.53322	0.131959
$136$	0	0
$137$	−18.5673	−1.58631	−0.793154	−	0.609021i	$-0.791563\pi$
$137$	−18.5673	−1.58631	−0.793154	+	0.609021i	$0.791563\pi$
$138$	0	0
$139$	15.0137	1.27345	0.636724	−	0.771092i	$-0.280289\pi$
$139$	15.0137	1.27345	0.636724	+	0.771092i	$0.280289\pi$
$140$	0	0
$141$	−6.81577	−0.573991
$142$	0	0
$143$	4.33039	0.362125
$144$	0	0
$145$	2.71064	0.225106
$146$	0	0
$147$	−6.16174	−0.508212
$148$	0	0
$149$	22.6712	1.85729	0.928647	−	0.370965i	$-0.120973\pi$
$149$	22.6712	1.85729	0.928647	+	0.370965i	$0.120973\pi$
$150$	0	0
$151$	19.2737	1.56847	0.784237	−	0.620462i	$-0.213055\pi$
$151$	19.2737	1.56847	0.784237	+	0.620462i	$0.213055\pi$
$152$	0	0
$153$	−1.50046	−0.121305
$154$	0	0
$155$	5.82614	0.467967
$156$	0	0
$157$	15.3895	1.22821	0.614107	−	0.789222i	$-0.289516\pi$
$157$	15.3895	1.22821	0.614107	+	0.789222i	$0.289516\pi$
$158$	0	0
$159$	1.43276	0.113625
$160$	0	0
$161$	−4.53809	−0.357651
$162$	0	0
$163$	−2.65971	−0.208325	−0.104162	−	0.994560i	$-0.533216\pi$
$163$	−2.65971	−0.208325	−0.104162	+	0.994560i	$0.533216\pi$
$164$	0	0
$165$	4.74943	0.369742
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.0457	−0.849672
$170$	0	0
$171$	−4.21086	−0.322013
$172$	0	0
$173$	16.5038	1.25476	0.627382	−	0.778712i	$-0.284126\pi$
$173$	16.5038	1.25476	0.627382	+	0.778712i	$0.284126\pi$
$174$	0	0
$175$	2.42554	0.183353
$176$	0	0
$177$	12.9348	0.972242
$178$	0	0
$179$	0.202161	0.0151102	0.00755512	−	0.999971i	$-0.497595\pi$
$179$	0.202161	0.0151102	0.00755512	+	0.999971i	$0.497595\pi$
$180$	0	0
$181$	−2.91030	−0.216321	−0.108160	−	0.994133i	$-0.534496\pi$
$181$	−2.91030	−0.216321	−0.108160	+	0.994133i	$0.534496\pi$
$182$	0	0
$183$	−3.76679	−0.278449
$184$	0	0
$185$	6.36965	0.468306
$186$	0	0
$187$	−4.64792	−0.339890
$188$	0	0
$189$	−0.915565	−0.0665975
$190$	0	0
$191$	−15.2680	−1.10475	−0.552377	−	0.833594i	$-0.686279\pi$
$191$	−15.2680	−1.10475	−0.552377	+	0.833594i	$0.686279\pi$
$192$	0	0
$193$	8.99644	0.647578	0.323789	−	0.946129i	$-0.395043\pi$
$193$	8.99644	0.647578	0.323789	+	0.946129i	$0.395043\pi$
$194$	0	0
$195$	2.14337	0.153490
$196$	0	0
$197$	11.6292	0.828547	0.414273	−	0.910152i	$-0.364036\pi$
$197$	11.6292	0.828547	0.414273	+	0.910152i	$0.364036\pi$
$198$	0	0
$199$	12.6305	0.895350	0.447675	−	0.894196i	$-0.352252\pi$
$199$	12.6305	0.895350	0.447675	+	0.894196i	$0.352252\pi$
$200$	0	0
$201$	7.30009	0.514909
$202$	0	0
$203$	−1.61866	−0.113607
$204$	0	0
$205$	12.7696	0.891867
$206$	0	0
$207$	4.95660	0.344508
$208$	0	0
$209$	−13.0439	−0.902264
$210$	0	0
$211$	−11.2769	−0.776331	−0.388166	−	0.921590i	$-0.626891\pi$
$211$	−11.2769	−0.776331	−0.388166	+	0.921590i	$0.626891\pi$
$212$	0	0
$213$	0.883952	0.0605674
$214$	0	0
$215$	12.9664	0.884303
$216$	0	0
$217$	−3.47908	−0.236175
$218$	0	0
$219$	−0.891880	−0.0602677
$220$	0	0
$221$	−2.09756	−0.141097
$222$	0	0
$223$	−11.4567	−0.767196	−0.383598	−	0.923500i	$-0.625315\pi$
$223$	−11.4567	−0.767196	−0.383598	+	0.923500i	$0.625315\pi$
$224$	0	0
$225$	−2.64923	−0.176615
$226$	0	0
$227$	6.99214	0.464085	0.232042	−	0.972706i	$-0.425459\pi$
$227$	6.99214	0.464085	0.232042	+	0.972706i	$0.425459\pi$
$228$	0	0
$229$	4.84467	0.320145	0.160073	−	0.987105i	$-0.448827\pi$
$229$	4.84467	0.320145	0.160073	+	0.987105i	$0.448827\pi$
$230$	0	0
$231$	−2.83612	−0.186603
$232$	0	0
$233$	−26.4019	−1.72965	−0.864824	−	0.502075i	$-0.832570\pi$
$233$	−26.4019	−1.72965	−0.864824	+	0.502075i	$0.832570\pi$
$234$	0	0
$235$	−10.4501	−0.681689
$236$	0	0
$237$	−1.12995	−0.0733979
$238$	0	0
$239$	23.9728	1.55067	0.775335	−	0.631550i	$-0.217581\pi$
$239$	23.9728	1.55067	0.775335	+	0.631550i	$0.217581\pi$
$240$	0	0
$241$	6.33066	0.407794	0.203897	−	0.978992i	$-0.434639\pi$
$241$	6.33066	0.407794	0.203897	+	0.978992i	$0.434639\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−9.44733	−0.603568
$246$	0	0
$247$	−5.88657	−0.374553
$248$	0	0
$249$	4.26402	0.270221
$250$	0	0
$251$	10.5048	0.663059	0.331529	−	0.943445i	$-0.392435\pi$
$251$	10.5048	0.663059	0.331529	+	0.943445i	$0.392435\pi$
$252$	0	0
$253$	15.3539	0.965294
$254$	0	0
$255$	−2.30053	−0.144065
$256$	0	0
$257$	−25.5749	−1.59532	−0.797661	−	0.603107i	$-0.793929\pi$
$257$	−25.5749	−1.59532	−0.797661	+	0.603107i	$0.793929\pi$
$258$	0	0
$259$	−3.80364	−0.236346
$260$	0	0
$261$	1.76793	0.109432
$262$	0	0
$263$	−20.5255	−1.26566	−0.632828	−	0.774292i	$-0.718106\pi$
$263$	−20.5255	−1.26566	−0.632828	+	0.774292i	$0.718106\pi$
$264$	0	0
$265$	2.19674	0.134944
$266$	0	0
$267$	17.6794	1.08196
$268$	0	0
$269$	11.2640	0.686781	0.343390	−	0.939193i	$-0.388425\pi$
$269$	11.2640	0.686781	0.343390	+	0.939193i	$0.388425\pi$
$270$	0	0
$271$	−5.25659	−0.319315	−0.159658	−	0.987172i	$-0.551039\pi$
$271$	−5.25659	−0.319315	−0.159658	+	0.987172i	$0.551039\pi$
$272$	0	0
$273$	−1.27991	−0.0774638
$274$	0	0
$275$	−8.20644	−0.494867
$276$	0	0
$277$	−10.4026	−0.625034	−0.312517	−	0.949912i	$-0.601172\pi$
$277$	−10.4026	−0.625034	−0.312517	+	0.949912i	$0.601172\pi$
$278$	0	0
$279$	3.79993	0.227496
$280$	0	0
$281$	−21.6130	−1.28932	−0.644661	−	0.764469i	$-0.723001\pi$
$281$	−21.6130	−1.28932	−0.644661	+	0.764469i	$0.723001\pi$
$282$	0	0
$283$	24.2913	1.44397	0.721983	−	0.691911i	$-0.243231\pi$
$283$	24.2913	1.44397	0.721983	+	0.691911i	$0.243231\pi$
$284$	0	0
$285$	−6.45620	−0.382432
$286$	0	0
$287$	−7.62537	−0.450111
$288$	0	0
$289$	−14.7486	−0.867567
$290$	0	0
$291$	−4.28751	−0.251339
$292$	0	0
$293$	15.1490	0.885014	0.442507	−	0.896765i	$-0.354089\pi$
$293$	15.1490	0.885014	0.442507	+	0.896765i	$0.354089\pi$
$294$	0	0
$295$	19.8320	1.15466
$296$	0	0
$297$	3.09767	0.179745
$298$	0	0
$299$	6.92907	0.400718
$300$	0	0
$301$	−7.74290	−0.446293
$302$	0	0
$303$	−15.5844	−0.895303
$304$	0	0
$305$	−5.77533	−0.330694
$306$	0	0
$307$	8.37234	0.477835	0.238917	−	0.971040i	$-0.423208\pi$
$307$	8.37234	0.477835	0.238917	+	0.971040i	$0.423208\pi$
$308$	0	0
$309$	−5.71458	−0.325091
$310$	0	0
$311$	4.65962	0.264223	0.132111	−	0.991235i	$-0.457824\pi$
$311$	4.65962	0.264223	0.132111	+	0.991235i	$0.457824\pi$
$312$	0	0
$313$	−8.10404	−0.458067	−0.229034	−	0.973419i	$-0.573557\pi$
$313$	−8.10404	−0.458067	−0.229034	+	0.973419i	$0.573557\pi$
$314$	0	0
$315$	−1.40377	−0.0790932
$316$	0	0
$317$	−12.1018	−0.679706	−0.339853	−	0.940479i	$-0.610377\pi$
$317$	−12.1018	−0.679706	−0.339853	+	0.940479i	$0.610377\pi$
$318$	0	0
$319$	5.47648	0.306624
$320$	0	0
$321$	−3.17924	−0.177448
$322$	0	0
$323$	6.31822	0.351555
$324$	0	0
$325$	−3.70348	−0.205432
$326$	0	0
$327$	−12.5475	−0.693881
$328$	0	0
$329$	6.24027	0.344037
$330$	0	0
$331$	−11.5342	−0.633979	−0.316989	−	0.948429i	$-0.602672\pi$
$331$	−11.5342	−0.633979	−0.316989	+	0.948429i	$0.602672\pi$
$332$	0	0
$333$	4.15442	0.227661
$334$	0	0
$335$	11.1927	0.611521
$336$	0	0
$337$	4.87706	0.265670	0.132835	−	0.991138i	$-0.457592\pi$
$337$	4.87706	0.265670	0.132835	+	0.991138i	$0.457592\pi$
$338$	0	0
$339$	2.50783	0.136206
$340$	0	0
$341$	11.7709	0.637432
$342$	0	0
$343$	12.0504	0.650662
$344$	0	0
$345$	7.59958	0.409148
$346$	0	0
$347$	−5.41332	−0.290602	−0.145301	−	0.989387i	$-0.546415\pi$
$347$	−5.41332	−0.290602	−0.145301	+	0.989387i	$0.546415\pi$
$348$	0	0
$349$	16.3896	0.877313	0.438656	−	0.898655i	$-0.355455\pi$
$349$	16.3896	0.877313	0.438656	+	0.898655i	$0.355455\pi$
$350$	0	0
$351$	1.39795	0.0746169
$352$	0	0
$353$	−33.9228	−1.80553	−0.902765	−	0.430133i	$-0.858467\pi$
$353$	−33.9228	−1.80553	−0.902765	+	0.430133i	$0.858467\pi$
$354$	0	0
$355$	1.35530	0.0719316
$356$	0	0
$357$	1.37376	0.0727073
$358$	0	0
$359$	30.9733	1.63471	0.817354	−	0.576135i	$-0.195440\pi$
$359$	30.9733	1.63471	0.817354	+	0.576135i	$0.195440\pi$
$360$	0	0
$361$	−1.26863	−0.0667699
$362$	0	0
$363$	−1.40441	−0.0737126
$364$	0	0
$365$	−1.36745	−0.0715757
$366$	0	0
$367$	−19.7551	−1.03121	−0.515603	−	0.856827i	$-0.672432\pi$
$367$	−19.7551	−1.03121	−0.515603	+	0.856827i	$0.672432\pi$
$368$	0	0
$369$	8.32859	0.433569
$370$	0	0
$371$	−1.31178	−0.0681043
$372$	0	0
$373$	9.12807	0.472634	0.236317	−	0.971676i	$-0.424060\pi$
$373$	9.12807	0.472634	0.236317	+	0.971676i	$0.424060\pi$
$374$	0	0
$375$	−11.7280	−0.605630
$376$	0	0
$377$	2.47148	0.127288
$378$	0	0
$379$	−13.9800	−0.718103	−0.359052	−	0.933318i	$-0.616900\pi$
$379$	−13.9800	−0.718103	−0.359052	+	0.933318i	$0.616900\pi$
$380$	0	0
$381$	17.3437	0.888543
$382$	0	0
$383$	−34.7750	−1.77692	−0.888459	−	0.458956i	$-0.848224\pi$
$383$	−34.7750	−1.77692	−0.888459	+	0.458956i	$0.848224\pi$
$384$	0	0
$385$	−4.34841	−0.221615
$386$	0	0
$387$	8.45697	0.429892
$388$	0	0
$389$	−16.7127	−0.847369	−0.423684	−	0.905810i	$-0.639263\pi$
$389$	−16.7127	−0.847369	−0.423684	+	0.905810i	$0.639263\pi$
$390$	0	0
$391$	−7.43716	−0.376113
$392$	0	0
$393$	−16.3850	−0.826513
$394$	0	0
$395$	−1.73246	−0.0871696
$396$	0	0
$397$	−3.95221	−0.198356	−0.0991780	−	0.995070i	$-0.531621\pi$
$397$	−3.95221	−0.198356	−0.0991780	+	0.995070i	$0.531621\pi$
$398$	0	0
$399$	3.85532	0.193007
$400$	0	0
$401$	−19.9460	−0.996055	−0.498028	−	0.867161i	$-0.665942\pi$
$401$	−19.9460	−0.996055	−0.498028	+	0.867161i	$0.665942\pi$
$402$	0	0
$403$	5.31210	0.264615
$404$	0	0
$405$	1.53322	0.0761865
$406$	0	0
$407$	12.8690	0.637894
$408$	0	0
$409$	−4.18390	−0.206881	−0.103440	−	0.994636i	$-0.532985\pi$
$409$	−4.18390	−0.206881	−0.103440	+	0.994636i	$0.532985\pi$
$410$	0	0
$411$	−18.5673	−0.915856
$412$	0	0
$413$	−11.8427	−0.582740
$414$	0	0
$415$	6.53769	0.320923
$416$	0	0
$417$	15.0137	0.735225
$418$	0	0
$419$	−39.5510	−1.93219	−0.966097	−	0.258179i	$-0.916877\pi$
$419$	−39.5510	−1.93219	−0.966097	+	0.258179i	$0.916877\pi$
$420$	0	0
$421$	−15.4917	−0.755019	−0.377510	−	0.926006i	$-0.623220\pi$
$421$	−15.4917	−0.755019	−0.377510	+	0.926006i	$0.623220\pi$
$422$	0	0
$423$	−6.81577	−0.331394
$424$	0	0
$425$	3.97505	0.192818
$426$	0	0
$427$	3.44874	0.166896
$428$	0	0
$429$	4.33039	0.209073
$430$	0	0
$431$	16.0476	0.772984	0.386492	−	0.922293i	$-0.373687\pi$
$431$	16.0476	0.772984	0.386492	+	0.922293i	$0.373687\pi$
$432$	0	0
$433$	21.5403	1.03516	0.517580	−	0.855635i	$-0.326833\pi$
$433$	21.5403	1.03516	0.517580	+	0.855635i	$0.326833\pi$
$434$	0	0
$435$	2.71064	0.129965
$436$	0	0
$437$	−20.8716	−0.998423
$438$	0	0
$439$	20.1392	0.961191	0.480595	−	0.876942i	$-0.340421\pi$
$439$	20.1392	0.961191	0.480595	+	0.876942i	$0.340421\pi$
$440$	0	0
$441$	−6.16174	−0.293416
$442$	0	0
$443$	17.2013	0.817257	0.408629	−	0.912701i	$-0.366007\pi$
$443$	17.2013	0.817257	0.408629	+	0.912701i	$0.366007\pi$
$444$	0	0
$445$	27.1065	1.28497
$446$	0	0
$447$	22.6712	1.07231
$448$	0	0
$449$	−28.7950	−1.35892	−0.679461	−	0.733712i	$-0.737786\pi$
$449$	−28.7950	−1.35892	−0.679461	+	0.733712i	$0.737786\pi$
$450$	0	0
$451$	25.7993	1.21484
$452$	0	0
$453$	19.2737	0.905559
$454$	0	0
$455$	−1.96239	−0.0919983
$456$	0	0
$457$	31.8010	1.48759	0.743794	−	0.668409i	$-0.233024\pi$
$457$	31.8010	1.48759	0.743794	+	0.668409i	$0.233024\pi$
$458$	0	0
$459$	−1.50046	−0.0700353
$460$	0	0
$461$	23.3828	1.08905	0.544524	−	0.838745i	$-0.316710\pi$
$461$	23.3828	1.08905	0.544524	+	0.838745i	$0.316710\pi$
$462$	0	0
$463$	39.4475	1.83328	0.916640	−	0.399714i	$-0.130891\pi$
$463$	39.4475	1.83328	0.916640	+	0.399714i	$0.130891\pi$
$464$	0	0
$465$	5.82614	0.270181
$466$	0	0
$467$	−31.6227	−1.46332	−0.731661	−	0.681668i	$-0.761255\pi$
$467$	−31.6227	−1.46332	−0.731661	+	0.681668i	$0.761255\pi$
$468$	0	0
$469$	−6.68370	−0.308625
$470$	0	0
$471$	15.3895	0.709110
$472$	0	0
$473$	26.1969	1.20454
$474$	0	0
$475$	11.1555	0.511851
$476$	0	0
$477$	1.43276	0.0656014
$478$	0	0
$479$	−13.4628	−0.615129	−0.307565	−	0.951527i	$-0.599514\pi$
$479$	−13.4628	−0.615129	−0.307565	+	0.951527i	$0.599514\pi$
$480$	0	0
$481$	5.80766	0.264806
$482$	0	0
$483$	−4.53809	−0.206490
$484$	0	0
$485$	−6.57372	−0.298497
$486$	0	0
$487$	34.2506	1.55204	0.776022	−	0.630706i	$-0.217235\pi$
$487$	34.2506	1.55204	0.776022	+	0.630706i	$0.217235\pi$
$488$	0	0
$489$	−2.65971	−0.120276
$490$	0	0
$491$	−16.2041	−0.731279	−0.365639	−	0.930757i	$-0.619150\pi$
$491$	−16.2041	−0.731279	−0.365639	+	0.930757i	$0.619150\pi$
$492$	0	0
$493$	−2.65271	−0.119472
$494$	0	0
$495$	4.74943	0.213471
$496$	0	0
$497$	−0.809315	−0.0363027
$498$	0	0
$499$	19.1460	0.857091	0.428545	−	0.903520i	$-0.359026\pi$
$499$	19.1460	0.857091	0.428545	+	0.903520i	$0.359026\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	8.19733	0.365501	0.182750	−	0.983159i	$-0.441500\pi$
$503$	8.19733	0.365501	0.182750	+	0.983159i	$0.441500\pi$
$504$	0	0
$505$	−23.8944	−1.06329
$506$	0	0
$507$	−11.0457	−0.490559
$508$	0	0
$509$	15.4558	0.685065	0.342533	−	0.939506i	$-0.388715\pi$
$509$	15.4558	0.685065	0.342533	+	0.939506i	$0.388715\pi$
$510$	0	0
$511$	0.816574	0.0361231
$512$	0	0
$513$	−4.21086	−0.185914
$514$	0	0
$515$	−8.76173	−0.386088
$516$	0	0
$517$	−21.1130	−0.928550
$518$	0	0
$519$	16.5038	0.724438
$520$	0	0
$521$	22.9773	1.00666	0.503328	−	0.864096i	$-0.332109\pi$
$521$	22.9773	1.00666	0.503328	+	0.864096i	$0.332109\pi$
$522$	0	0
$523$	−1.86529	−0.0815636	−0.0407818	−	0.999168i	$-0.512985\pi$
$523$	−1.86529	−0.0815636	−0.0407818	+	0.999168i	$0.512985\pi$
$524$	0	0
$525$	2.42554	0.105859
$526$	0	0
$527$	−5.70163	−0.248367
$528$	0	0
$529$	1.56789	0.0681692
$530$	0	0
$531$	12.9348	0.561324
$532$	0	0
$533$	11.6429	0.504312
$534$	0	0
$535$	−4.87449	−0.210743
$536$	0	0
$537$	0.202161	0.00872390
$538$	0	0
$539$	−19.0871	−0.822138
$540$	0	0
$541$	−35.4871	−1.52571	−0.762854	−	0.646570i	$-0.776203\pi$
$541$	−35.4871	−1.52571	−0.762854	+	0.646570i	$0.776203\pi$
$542$	0	0
$543$	−2.91030	−0.124893
$544$	0	0
$545$	−19.2382	−0.824074
$546$	0	0
$547$	1.16371	0.0497568	0.0248784	−	0.999690i	$-0.492080\pi$
$547$	1.16371	0.0497568	0.0248784	+	0.999690i	$0.492080\pi$
$548$	0	0
$549$	−3.76679	−0.160763
$550$	0	0
$551$	−7.44452	−0.317147
$552$	0	0
$553$	1.03454	0.0439931
$554$	0	0
$555$	6.36965	0.270377
$556$	0	0
$557$	−27.6032	−1.16958	−0.584792	−	0.811183i	$-0.698824\pi$
$557$	−27.6032	−1.16958	−0.584792	+	0.811183i	$0.698824\pi$
$558$	0	0
$559$	11.8224	0.500034
$560$	0	0
$561$	−4.64792	−0.196235
$562$	0	0
$563$	26.7132	1.12583	0.562913	−	0.826516i	$-0.309681\pi$
$563$	26.7132	1.12583	0.562913	+	0.826516i	$0.309681\pi$
$564$	0	0
$565$	3.84506	0.161763
$566$	0	0
$567$	−0.915565	−0.0384501
$568$	0	0
$569$	−24.6239	−1.03229	−0.516143	−	0.856502i	$-0.672633\pi$
$569$	−24.6239	−1.03229	−0.516143	+	0.856502i	$0.672633\pi$
$570$	0	0
$571$	−16.6623	−0.697294	−0.348647	−	0.937254i	$-0.613359\pi$
$571$	−16.6623	−0.697294	−0.348647	+	0.937254i	$0.613359\pi$
$572$	0	0
$573$	−15.2680	−0.637830
$574$	0	0
$575$	−13.1312	−0.547607
$576$	0	0
$577$	−10.9442	−0.455612	−0.227806	−	0.973707i	$-0.573155\pi$
$577$	−10.9442	−0.455612	−0.227806	+	0.973707i	$0.573155\pi$
$578$	0	0
$579$	8.99644	0.373879
$580$	0	0
$581$	−3.90398	−0.161965
$582$	0	0
$583$	4.43821	0.183812
$584$	0	0
$585$	2.14337	0.0886173
$586$	0	0
$587$	1.12881	0.0465908	0.0232954	−	0.999729i	$-0.492584\pi$
$587$	1.12881	0.0465908	0.0232954	+	0.999729i	$0.492584\pi$
$588$	0	0
$589$	−16.0010	−0.659309
$590$	0	0
$591$	11.6292	0.478362
$592$	0	0
$593$	21.3857	0.878205	0.439102	−	0.898437i	$-0.355297\pi$
$593$	21.3857	0.878205	0.439102	+	0.898437i	$0.355297\pi$
$594$	0	0
$595$	2.10629	0.0863494
$596$	0	0
$597$	12.6305	0.516930
$598$	0	0
$599$	−13.0654	−0.533837	−0.266919	−	0.963719i	$-0.586006\pi$
$599$	−13.0654	−0.533837	−0.266919	+	0.963719i	$0.586006\pi$
$600$	0	0
$601$	−7.93394	−0.323632	−0.161816	−	0.986821i	$-0.551735\pi$
$601$	−7.93394	−0.323632	−0.161816	+	0.986821i	$0.551735\pi$
$602$	0	0
$603$	7.30009	0.297283
$604$	0	0
$605$	−2.15328	−0.0875433
$606$	0	0
$607$	35.6021	1.44504	0.722522	−	0.691348i	$-0.242983\pi$
$607$	35.6021	1.44504	0.722522	+	0.691348i	$0.242983\pi$
$608$	0	0
$609$	−1.61866	−0.0655913
$610$	0	0
$611$	−9.52809	−0.385465
$612$	0	0
$613$	9.26461	0.374194	0.187097	−	0.982341i	$-0.440092\pi$
$613$	9.26461	0.374194	0.187097	+	0.982341i	$0.440092\pi$
$614$	0	0
$615$	12.7696	0.514920
$616$	0	0
$617$	−18.9004	−0.760902	−0.380451	−	0.924801i	$-0.624231\pi$
$617$	−18.9004	−0.760902	−0.380451	+	0.924801i	$0.624231\pi$
$618$	0	0
$619$	−1.98374	−0.0797333	−0.0398666	−	0.999205i	$-0.512693\pi$
$619$	−1.98374	−0.0797333	−0.0398666	+	0.999205i	$0.512693\pi$
$620$	0	0
$621$	4.95660	0.198902
$622$	0	0
$623$	−16.1866	−0.648504
$624$	0	0
$625$	−4.73548	−0.189419
$626$	0	0
$627$	−13.0439	−0.520923
$628$	0	0
$629$	−6.23352	−0.248547
$630$	0	0
$631$	−28.1079	−1.11896	−0.559480	−	0.828844i	$-0.688999\pi$
$631$	−28.1079	−1.11896	−0.559480	+	0.828844i	$0.688999\pi$
$632$	0	0
$633$	−11.2769	−0.448215
$634$	0	0
$635$	26.5917	1.05526
$636$	0	0
$637$	−8.61379	−0.341291
$638$	0	0
$639$	0.883952	0.0349686
$640$	0	0
$641$	−13.6872	−0.540611	−0.270306	−	0.962775i	$-0.587125\pi$
$641$	−13.6872	−0.540611	−0.270306	+	0.962775i	$0.587125\pi$
$642$	0	0
$643$	11.8120	0.465820	0.232910	−	0.972498i	$-0.425175\pi$
$643$	11.8120	0.465820	0.232910	+	0.972498i	$0.425175\pi$
$644$	0	0
$645$	12.9664	0.510553
$646$	0	0
$647$	12.9647	0.509696	0.254848	−	0.966981i	$-0.417975\pi$
$647$	12.9647	0.509696	0.254848	+	0.966981i	$0.417975\pi$
$648$	0	0
$649$	40.0679	1.57280
$650$	0	0
$651$	−3.47908	−0.136356
$652$	0	0
$653$	−47.8071	−1.87084	−0.935419	−	0.353541i	$-0.884978\pi$
$653$	−47.8071	−1.87084	−0.935419	+	0.353541i	$0.884978\pi$
$654$	0	0
$655$	−25.1219	−0.981592
$656$	0	0
$657$	−0.891880	−0.0347956
$658$	0	0
$659$	16.2125	0.631550	0.315775	−	0.948834i	$-0.397735\pi$
$659$	16.2125	0.631550	0.315775	+	0.948834i	$0.397735\pi$
$660$	0	0
$661$	−41.9378	−1.63119	−0.815595	−	0.578623i	$-0.803590\pi$
$661$	−41.9378	−1.63119	−0.815595	+	0.578623i	$0.803590\pi$
$662$	0	0
$663$	−2.09756	−0.0814625
$664$	0	0
$665$	5.91106	0.229221
$666$	0	0
$667$	8.76294	0.339302
$668$	0	0
$669$	−11.4567	−0.442941
$670$	0	0
$671$	−11.6683	−0.450449
$672$	0	0
$673$	−17.9537	−0.692064	−0.346032	−	0.938223i	$-0.612471\pi$
$673$	−17.9537	−0.692064	−0.346032	+	0.938223i	$0.612471\pi$
$674$	0	0
$675$	−2.64923	−0.101969
$676$	0	0
$677$	−9.80016	−0.376651	−0.188325	−	0.982107i	$-0.560306\pi$
$677$	−9.80016	−0.376651	−0.188325	+	0.982107i	$0.560306\pi$
$678$	0	0
$679$	3.92550	0.150647
$680$	0	0
$681$	6.99214	0.267939
$682$	0	0
$683$	−42.7950	−1.63751	−0.818753	−	0.574145i	$-0.805334\pi$
$683$	−42.7950	−1.63751	−0.818753	+	0.574145i	$0.805334\pi$
$684$	0	0
$685$	−28.4678	−1.08770
$686$	0	0
$687$	4.84467	0.184836
$688$	0	0
$689$	2.00292	0.0763051
$690$	0	0
$691$	47.2115	1.79601	0.898005	−	0.439985i	$-0.145016\pi$
$691$	47.2115	1.79601	0.898005	+	0.439985i	$0.145016\pi$
$692$	0	0
$693$	−2.83612	−0.107735
$694$	0	0
$695$	23.0194	0.873176
$696$	0	0
$697$	−12.4967	−0.473346
$698$	0	0
$699$	−26.4019	−0.998613
$700$	0	0
$701$	−0.0922506	−0.00348426	−0.00174213	−	0.999998i	$-0.500555\pi$
$701$	−0.0922506	−0.00348426	−0.00174213	+	0.999998i	$0.500555\pi$
$702$	0	0
$703$	−17.4937	−0.659787
$704$	0	0
$705$	−10.4501	−0.393573
$706$	0	0
$707$	14.2686	0.536624
$708$	0	0
$709$	−33.0225	−1.24019	−0.620093	−	0.784528i	$-0.712905\pi$
$709$	−33.0225	−1.24019	−0.620093	+	0.784528i	$0.712905\pi$
$710$	0	0
$711$	−1.12995	−0.0423763
$712$	0	0
$713$	18.8347	0.705366
$714$	0	0
$715$	6.63945	0.248301
$716$	0	0
$717$	23.9728	0.895279
$718$	0	0
$719$	−2.96188	−0.110459	−0.0552297	−	0.998474i	$-0.517589\pi$
$719$	−2.96188	−0.110459	−0.0552297	+	0.998474i	$0.517589\pi$
$720$	0	0
$721$	5.23207	0.194852
$722$	0	0
$723$	6.33066	0.235440
$724$	0	0
$725$	−4.68365	−0.173946
$726$	0	0
$727$	14.9351	0.553914	0.276957	−	0.960882i	$-0.410674\pi$
$727$	14.9351	0.553914	0.276957	+	0.960882i	$0.410674\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.6893	−0.469331
$732$	0	0
$733$	−28.0289	−1.03527	−0.517635	−	0.855602i	$-0.673187\pi$
$733$	−28.0289	−1.03527	−0.517635	+	0.855602i	$0.673187\pi$
$734$	0	0
$735$	−9.44733	−0.348470
$736$	0	0
$737$	22.6133	0.832971
$738$	0	0
$739$	−34.2689	−1.26060	−0.630300	−	0.776351i	$-0.717068\pi$
$739$	−34.2689	−1.26060	−0.630300	+	0.776351i	$0.717068\pi$
$740$	0	0
$741$	−5.88657	−0.216248
$742$	0	0
$743$	−40.7965	−1.49668	−0.748340	−	0.663316i	$-0.769149\pi$
$743$	−40.7965	−1.49668	−0.748340	+	0.663316i	$0.769149\pi$
$744$	0	0
$745$	34.7600	1.27351
$746$	0	0
$747$	4.26402	0.156012
$748$	0	0
$749$	2.91080	0.106358
$750$	0	0
$751$	41.0708	1.49869	0.749347	−	0.662177i	$-0.230367\pi$
$751$	41.0708	1.49869	0.749347	+	0.662177i	$0.230367\pi$
$752$	0	0
$753$	10.5048	0.382817
$754$	0	0
$755$	29.5509	1.07547
$756$	0	0
$757$	−33.5015	−1.21763	−0.608817	−	0.793311i	$-0.708356\pi$
$757$	−33.5015	−1.21763	−0.608817	+	0.793311i	$0.708356\pi$
$758$	0	0
$759$	15.3539	0.557313
$760$	0	0
$761$	−15.9152	−0.576926	−0.288463	−	0.957491i	$-0.593144\pi$
$761$	−15.9152	−0.576926	−0.288463	+	0.957491i	$0.593144\pi$
$762$	0	0
$763$	11.4881	0.415897
$764$	0	0
$765$	−2.30053	−0.0831760
$766$	0	0
$767$	18.0822	0.652912
$768$	0	0
$769$	−24.5047	−0.883661	−0.441830	−	0.897099i	$-0.645671\pi$
$769$	−24.5047	−0.883661	−0.441830	+	0.897099i	$0.645671\pi$
$770$	0	0
$771$	−25.5749	−0.921059
$772$	0	0
$773$	28.0247	1.00798	0.503990	−	0.863710i	$-0.331865\pi$
$773$	28.0247	1.00798	0.503990	+	0.863710i	$0.331865\pi$
$774$	0	0
$775$	−10.0669	−0.361612
$776$	0	0
$777$	−3.80364	−0.136455
$778$	0	0
$779$	−35.0706	−1.25653
$780$	0	0
$781$	2.73819	0.0979803
$782$	0	0
$783$	1.76793	0.0631808
$784$	0	0
$785$	23.5955	0.842160
$786$	0	0
$787$	−13.5173	−0.481840	−0.240920	−	0.970545i	$-0.577449\pi$
$787$	−13.5173	−0.481840	−0.240920	+	0.970545i	$0.577449\pi$
$788$	0	0
$789$	−20.5255	−0.730727
$790$	0	0
$791$	−2.29608	−0.0816391
$792$	0	0
$793$	−5.26577	−0.186993
$794$	0	0
$795$	2.19674	0.0779102
$796$	0	0
$797$	12.3513	0.437504	0.218752	−	0.975780i	$-0.429801\pi$
$797$	12.3513	0.437504	0.218752	+	0.975780i	$0.429801\pi$
$798$	0	0
$799$	10.2268	0.361797
$800$	0	0
$801$	17.6794	0.624671
$802$	0	0
$803$	−2.76275	−0.0974955
$804$	0	0
$805$	−6.95790	−0.245234
$806$	0	0
$807$	11.2640	0.396513
$808$	0	0
$809$	−49.9778	−1.75713	−0.878563	−	0.477626i	$-0.841497\pi$
$809$	−49.9778	−1.75713	−0.878563	+	0.477626i	$0.841497\pi$
$810$	0	0
$811$	−41.8896	−1.47094	−0.735472	−	0.677556i	$-0.763039\pi$
$811$	−41.8896	−1.47094	−0.735472	+	0.677556i	$0.763039\pi$
$812$	0	0
$813$	−5.25659	−0.184357
$814$	0	0
$815$	−4.07793	−0.142844
$816$	0	0
$817$	−35.6111	−1.24588
$818$	0	0
$819$	−1.27991	−0.0447237
$820$	0	0
$821$	−20.6369	−0.720232	−0.360116	−	0.932908i	$-0.617263\pi$
$821$	−20.6369	−0.720232	−0.360116	+	0.932908i	$0.617263\pi$
$822$	0	0
$823$	33.0136	1.15078	0.575390	−	0.817879i	$-0.304850\pi$
$823$	33.0136	1.15078	0.575390	+	0.817879i	$0.304850\pi$
$824$	0	0
$825$	−8.20644	−0.285711
$826$	0	0
$827$	−4.76225	−0.165600	−0.0827998	−	0.996566i	$-0.526386\pi$
$827$	−4.76225	−0.165600	−0.0827998	+	0.996566i	$0.526386\pi$
$828$	0	0
$829$	37.3978	1.29888	0.649439	−	0.760414i	$-0.275004\pi$
$829$	37.3978	1.29888	0.649439	+	0.760414i	$0.275004\pi$
$830$	0	0
$831$	−10.4026	−0.360863
$832$	0	0
$833$	9.24542	0.320335
$834$	0	0
$835$	1.53322	0.0530594
$836$	0	0
$837$	3.79993	0.131345
$838$	0	0
$839$	−29.8684	−1.03117	−0.515585	−	0.856838i	$-0.672425\pi$
$839$	−29.8684	−1.03117	−0.515585	+	0.856838i	$0.672425\pi$
$840$	0	0
$841$	−25.8744	−0.892221
$842$	0	0
$843$	−21.6130	−0.744390
$844$	0	0
$845$	−16.9356	−0.582602
$846$	0	0
$847$	1.28583	0.0441817
$848$	0	0
$849$	24.2913	0.833674
$850$	0	0
$851$	20.5918	0.705877
$852$	0	0
$853$	21.5896	0.739214	0.369607	−	0.929188i	$-0.379492\pi$
$853$	21.5896	0.739214	0.369607	+	0.929188i	$0.379492\pi$
$854$	0	0
$855$	−6.45620	−0.220797
$856$	0	0
$857$	−30.7368	−1.04995	−0.524975	−	0.851118i	$-0.675925\pi$
$857$	−30.7368	−1.04995	−0.524975	+	0.851118i	$0.675925\pi$
$858$	0	0
$859$	−0.952468	−0.0324978	−0.0162489	−	0.999868i	$-0.505172\pi$
$859$	−0.952468	−0.0324978	−0.0162489	+	0.999868i	$0.505172\pi$
$860$	0	0
$861$	−7.62537	−0.259872
$862$	0	0
$863$	−14.7115	−0.500786	−0.250393	−	0.968144i	$-0.580560\pi$
$863$	−14.7115	−0.500786	−0.250393	+	0.968144i	$0.580560\pi$
$864$	0	0
$865$	25.3041	0.860365
$866$	0	0
$867$	−14.7486	−0.500890
$868$	0	0
$869$	−3.50021	−0.118736
$870$	0	0
$871$	10.2051	0.345788
$872$	0	0
$873$	−4.28751	−0.145110
$874$	0	0
$875$	10.7377	0.363001
$876$	0	0
$877$	−19.3949	−0.654919	−0.327459	−	0.944865i	$-0.606192\pi$
$877$	−19.3949	−0.654919	−0.327459	+	0.944865i	$0.606192\pi$
$878$	0	0
$879$	15.1490	0.510963
$880$	0	0
$881$	−40.8536	−1.37639	−0.688197	−	0.725524i	$-0.741597\pi$
$881$	−40.8536	−1.37639	−0.688197	+	0.725524i	$0.741597\pi$
$882$	0	0
$883$	−51.0635	−1.71843	−0.859213	−	0.511619i	$-0.829046\pi$
$883$	−51.0635	−1.71843	−0.859213	+	0.511619i	$0.829046\pi$
$884$	0	0
$885$	19.8320	0.666646
$886$	0	0
$887$	−0.844140	−0.0283434	−0.0141717	−	0.999900i	$-0.504511\pi$
$887$	−0.844140	−0.0283434	−0.0141717	+	0.999900i	$0.504511\pi$
$888$	0	0
$889$	−15.8792	−0.532573
$890$	0	0
$891$	3.09767	0.103776
$892$	0	0
$893$	28.7003	0.960418
$894$	0	0
$895$	0.309958	0.0103608
$896$	0	0
$897$	6.92907	0.231355
$898$	0	0
$899$	6.71802	0.224058
$900$	0	0
$901$	−2.14979	−0.0716198
$902$	0	0
$903$	−7.74290	−0.257668
$904$	0	0
$905$	−4.46214	−0.148326
$906$	0	0
$907$	14.2678	0.473754	0.236877	−	0.971540i	$-0.423876\pi$
$907$	14.2678	0.473754	0.236877	+	0.971540i	$0.423876\pi$
$908$	0	0
$909$	−15.5844	−0.516903
$910$	0	0
$911$	2.15685	0.0714598	0.0357299	−	0.999361i	$-0.488624\pi$
$911$	2.15685	0.0714598	0.0357299	+	0.999361i	$0.488624\pi$
$912$	0	0
$913$	13.2085	0.437139
$914$	0	0
$915$	−5.77533	−0.190926
$916$	0	0
$917$	15.0015	0.495394
$918$	0	0
$919$	−50.7401	−1.67376	−0.836881	−	0.547386i	$-0.815623\pi$
$919$	−50.7401	−1.67376	−0.836881	+	0.547386i	$0.815623\pi$
$920$	0	0
$921$	8.37234	0.275878
$922$	0	0
$923$	1.23572	0.0406742
$924$	0	0
$925$	−11.0060	−0.361874
$926$	0	0
$927$	−5.71458	−0.187692
$928$	0	0
$929$	−20.0006	−0.656199	−0.328100	−	0.944643i	$-0.606408\pi$
$929$	−20.0006	−0.656199	−0.328100	+	0.944643i	$0.606408\pi$
$930$	0	0
$931$	25.9463	0.850354
$932$	0	0
$933$	4.65962	0.152549
$934$	0	0
$935$	−7.12631	−0.233055
$936$	0	0
$937$	8.79635	0.287364	0.143682	−	0.989624i	$-0.454106\pi$
$937$	8.79635	0.287364	0.143682	+	0.989624i	$0.454106\pi$
$938$	0	0
$939$	−8.10404	−0.264465
$940$	0	0
$941$	−15.3968	−0.501921	−0.250961	−	0.967997i	$-0.580746\pi$
$941$	−15.3968	−0.501921	−0.250961	+	0.967997i	$0.580746\pi$
$942$	0	0
$943$	41.2815	1.34431
$944$	0	0
$945$	−1.40377	−0.0456645
$946$	0	0
$947$	−33.6342	−1.09297	−0.546483	−	0.837470i	$-0.684034\pi$
$947$	−33.6342	−1.09297	−0.546483	+	0.837470i	$0.684034\pi$
$948$	0	0
$949$	−1.24680	−0.0404729
$950$	0	0
$951$	−12.1018	−0.392429
$952$	0	0
$953$	19.0383	0.616712	0.308356	−	0.951271i	$-0.400221\pi$
$953$	19.0383	0.616712	0.308356	+	0.951271i	$0.400221\pi$
$954$	0	0
$955$	−23.4093	−0.757507
$956$	0	0
$957$	5.47648	0.177029
$958$	0	0
$959$	16.9995	0.548943
$960$	0	0
$961$	−16.5605	−0.534211
$962$	0	0
$963$	−3.17924	−0.102450
$964$	0	0
$965$	13.7935	0.444030
$966$	0	0
$967$	−37.3295	−1.20044	−0.600218	−	0.799836i	$-0.704920\pi$
$967$	−37.3295	−1.20044	−0.600218	+	0.799836i	$0.704920\pi$
$968$	0	0
$969$	6.31822	0.202970
$970$	0	0
$971$	−52.8286	−1.69535	−0.847675	−	0.530516i	$-0.821998\pi$
$971$	−52.8286	−1.69535	−0.847675	+	0.530516i	$0.821998\pi$
$972$	0	0
$973$	−13.7460	−0.440678
$974$	0	0
$975$	−3.70348	−0.118606
$976$	0	0
$977$	31.0861	0.994532	0.497266	−	0.867598i	$-0.334337\pi$
$977$	31.0861	0.994532	0.497266	+	0.867598i	$0.334337\pi$
$978$	0	0
$979$	54.7651	1.75030
$980$	0	0
$981$	−12.5475	−0.400612
$982$	0	0
$983$	−2.74694	−0.0876136	−0.0438068	−	0.999040i	$-0.513949\pi$
$983$	−2.74694	−0.0876136	−0.0438068	+	0.999040i	$0.513949\pi$
$984$	0	0
$985$	17.8302	0.568117
$986$	0	0
$987$	6.24027	0.198630
$988$	0	0
$989$	41.9178	1.33291
$990$	0	0
$991$	36.1120	1.14713	0.573567	−	0.819158i	$-0.305559\pi$
$991$	36.1120	1.14713	0.573567	+	0.819158i	$0.305559\pi$
$992$	0	0
$993$	−11.5342	−0.366028
$994$	0	0
$995$	19.3653	0.613922
$996$	0	0
$997$	−52.4769	−1.66196	−0.830980	−	0.556302i	$-0.812220\pi$
$997$	−52.4769	−1.66196	−0.830980	+	0.556302i	$0.812220\pi$
$998$	0	0
$999$	4.15442	0.131440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.7	✓	12
4.3	odd	2		8016.2.a.bf.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.7	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.7		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.53322 q^{5} -0.915565 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.53322 q^{5} -0.915565 q^{7} +1.00000 q^{9} +3.09767 q^{11} +1.39795 q^{13} +1.53322 q^{15} -1.50046 q^{17} -4.21086 q^{19} -0.915565 q^{21} +4.95660 q^{23} -2.64923 q^{25} +1.00000 q^{27} +1.76793 q^{29} +3.79993 q^{31} +3.09767 q^{33} -1.40377 q^{35} +4.15442 q^{37} +1.39795 q^{39} +8.32859 q^{41} +8.45697 q^{43} +1.53322 q^{45} -6.81577 q^{47} -6.16174 q^{49} -1.50046 q^{51} +1.43276 q^{53} +4.74943 q^{55} -4.21086 q^{57} +12.9348 q^{59} -3.76679 q^{61} -0.915565 q^{63} +2.14337 q^{65} +7.30009 q^{67} +4.95660 q^{69} +0.883952 q^{71} -0.891880 q^{73} -2.64923 q^{75} -2.83612 q^{77} -1.12995 q^{79} +1.00000 q^{81} +4.26402 q^{83} -2.30053 q^{85} +1.76793 q^{87} +17.6794 q^{89} -1.27991 q^{91} +3.79993 q^{93} -6.45620 q^{95} -4.28751 q^{97} +3.09767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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