LMFDB - Embedded newform 4000.2.c.f.1249.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4000,2,Mod(1249,4000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-8,0,-26,0,0,0,0,0,0,0,18,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$31.9401608085$
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} + 22x^{10} + 179x^{8} + 646x^{6} + 929x^{4} + 252x^{2} + 16 $ x^12 + 22x^10 + 179x^8 + 646x^6 + 929x^4 + 252*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.9
Root			$1.81030i$ of defining polynomial
Character	$\chi$	$=$	4000.1249
Dual form			4000.2.c.f.1249.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81030i q^{3} -4.37760i q^{7} -0.277175 q^{9} -5.82433 q^{11} -5.34625i q^{13} +5.41151i q^{17} -1.59789 q^{19} +7.92476 q^{21} -0.340859i q^{23} +4.92912i q^{27} +4.96428 q^{29} +4.63796 q^{31} -10.5438i q^{33} +8.40977i q^{37} +9.67829 q^{39} -2.34244 q^{41} -7.10407i q^{43} -0.879427i q^{47} -12.1634 q^{49} -9.79645 q^{51} +1.30241i q^{53} -2.89265i q^{57} -11.0726 q^{59} -14.4423 q^{61} +1.21336i q^{63} +3.01420i q^{67} +0.617056 q^{69} -9.37319 q^{71} +13.7157i q^{73} +25.4966i q^{77} -5.85098 q^{79} -9.75470 q^{81} +8.54562i q^{83} +8.98682i q^{87} +18.0227 q^{89} -23.4037 q^{91} +8.39608i q^{93} -10.0919i q^{97} +1.61436 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9} - 26 q^{11} + 18 q^{19} - 4 q^{29} - 24 q^{31} + 52 q^{39} - 10 q^{41} + 6 q^{49} - 36 q^{51} + 50 q^{59} - 12 q^{61} + 24 q^{69} - 68 q^{71} + 32 q^{79} - 36 q^{81} + 6 q^{89} - 52 q^{91}+ \cdots + 86 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 - 26 * q^11 + 18 * q^19 - 4 * q^29 - 24 * q^31 + 52 * q^39 - 10 * q^41 + 6 * q^49 - 36 * q^51 + 50 * q^59 - 12 * q^61 + 24 * q^69 - 68 * q^71 + 32 * q^79 - 36 * q^81 + 6 * q^89 - 52 * q^91 + 86 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.81030i	1.04518i	0.852586	+	0.522588i	$0.175033\pi$
$3$	1.81030i	1.04518i	−0.852586	+	0.522588i	$0.824967\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.37760i	− 1.65458i	−0.561777	−	0.827289i	$-0.689882\pi$
$7$	− 4.37760i	− 1.65458i	0.561777	−	0.827289i	$-0.310118\pi$
$8$	0	0
$9$	−0.277175	−0.0923916
$10$	0	0
$11$	−5.82433	−1.75610	−0.878051	−	0.478567i	$-0.841156\pi$
$11$	−5.82433	−1.75610	−0.878051	+	0.478567i	$0.841156\pi$
$12$	0	0
$13$	− 5.34625i	− 1.48278i	−0.671073	−	0.741391i	$-0.734166\pi$
$13$	− 5.34625i	− 1.48278i	0.671073	−	0.741391i	$-0.265834\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.41151i	1.31248i	0.754550	+	0.656242i	$0.227855\pi$
$17$	5.41151i	1.31248i	−0.754550	+	0.656242i	$0.772145\pi$
$18$	0	0
$19$	−1.59789	−0.366580	−0.183290	−	0.983059i	$-0.558675\pi$
$19$	−1.59789	−0.366580	−0.183290	+	0.983059i	$0.558675\pi$
$20$	0	0
$21$	7.92476	1.72932
$22$	0	0
$23$	− 0.340859i	− 0.0710740i	−0.999368	−	0.0355370i	$-0.988686\pi$
$23$	− 0.340859i	− 0.0710740i	0.999368	−	0.0355370i	$-0.0113142\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.92912i	0.948610i
$28$	0	0
$29$	4.96428	0.921844	0.460922	−	0.887441i	$-0.347519\pi$
$29$	4.96428	0.921844	0.460922	+	0.887441i	$0.347519\pi$
$30$	0	0
$31$	4.63796	0.833002	0.416501	−	0.909135i	$-0.363256\pi$
$31$	4.63796	0.833002	0.416501	+	0.909135i	$0.363256\pi$
$32$	0	0
$33$	− 10.5438i	− 1.83543i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.40977i	1.38256i	0.722588	+	0.691278i	$0.242952\pi$
$37$	8.40977i	1.38256i	−0.722588	+	0.691278i	$0.757048\pi$
$38$	0	0
$39$	9.67829	1.54977
$40$	0	0
$41$	−2.34244	−0.365828	−0.182914	−	0.983129i	$-0.558553\pi$
$41$	−2.34244	−0.365828	−0.182914	+	0.983129i	$0.558553\pi$
$42$	0	0
$43$	− 7.10407i	− 1.08336i	−0.840585	−	0.541680i	$-0.817788\pi$
$43$	− 7.10407i	− 1.08336i	0.840585	−	0.541680i	$-0.182212\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.879427i	− 0.128278i	−0.997941	−	0.0641388i	$-0.979570\pi$
$47$	− 0.879427i	− 0.128278i	0.997941	−	0.0641388i	$-0.0204300\pi$
$48$	0	0
$49$	−12.1634	−1.73763
$50$	0	0
$51$	−9.79645	−1.37178
$52$	0	0
$53$	1.30241i	0.178900i	0.995991	+	0.0894502i	$0.0285110\pi$
$53$	1.30241i	0.178900i	−0.995991	+	0.0894502i	$0.971489\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.89265i	− 0.383141i
$58$	0	0
$59$	−11.0726	−1.44153	−0.720767	−	0.693178i	$-0.756210\pi$
$59$	−11.0726	−1.44153	−0.720767	+	0.693178i	$0.756210\pi$
$60$	0	0
$61$	−14.4423	−1.84915	−0.924574	−	0.381003i	$-0.875579\pi$
$61$	−14.4423	−1.84915	−0.924574	+	0.381003i	$0.875579\pi$
$62$	0	0
$63$	1.21336i	0.152869i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.01420i	0.368243i	0.982903	+	0.184122i	$0.0589440\pi$
$67$	3.01420i	0.368243i	−0.982903	+	0.184122i	$0.941056\pi$
$68$	0	0
$69$	0.617056	0.0742848
$70$	0	0
$71$	−9.37319	−1.11239	−0.556197	−	0.831051i	$-0.687740\pi$
$71$	−9.37319	−1.11239	−0.556197	+	0.831051i	$0.687740\pi$
$72$	0	0
$73$	13.7157i	1.60530i	0.596451	+	0.802649i	$0.296577\pi$
$73$	13.7157i	1.60530i	−0.596451	+	0.802649i	$0.703423\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	25.4966i	2.90561i
$78$	0	0
$79$	−5.85098	−0.658287	−0.329143	−	0.944280i	$-0.606760\pi$
$79$	−5.85098	−0.658287	−0.329143	+	0.944280i	$0.606760\pi$
$80$	0	0
$81$	−9.75470	−1.08386
$82$	0	0
$83$	8.54562i	0.938003i	0.883197	+	0.469002i	$0.155386\pi$
$83$	8.54562i	0.938003i	−0.883197	+	0.469002i	$0.844614\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.98682i	0.963488i
$88$	0	0
$89$	18.0227	1.91040	0.955202	−	0.295956i	$-0.0956381\pi$
$89$	18.0227	1.91040	0.955202	+	0.295956i	$0.0956381\pi$
$90$	0	0
$91$	−23.4037	−2.45338
$92$	0	0
$93$	8.39608i	0.870633i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0919i	− 1.02468i	−0.858783	−	0.512340i	$-0.828779\pi$
$97$	− 10.0919i	− 1.02468i	0.858783	−	0.512340i	$-0.171221\pi$
$98$	0	0
$99$	1.61436	0.162249
$100$	0	0
$101$	−6.64409	−0.661111	−0.330556	−	0.943787i	$-0.607236\pi$
$101$	−6.64409	−0.661111	−0.330556	+	0.943787i	$0.607236\pi$
$102$	0	0
$103$	− 9.84923i	− 0.970474i	−0.874383	−	0.485237i	$-0.838733\pi$
$103$	− 9.84923i	− 0.970474i	0.874383	−	0.485237i	$-0.161267\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.941408i	0.0910093i	0.998964	+	0.0455047i	$0.0144896\pi$
$107$	0.941408i	0.0910093i	−0.998964	+	0.0455047i	$0.985510\pi$
$108$	0	0
$109$	−10.2276	−0.979627	−0.489813	−	0.871827i	$-0.662935\pi$
$109$	−10.2276	−0.979627	−0.489813	+	0.871827i	$0.662935\pi$
$110$	0	0
$111$	−15.2242	−1.44501
$112$	0	0
$113$	8.90474i	0.837687i	0.908058	+	0.418844i	$0.137564\pi$
$113$	8.90474i	0.837687i	−0.908058	+	0.418844i	$0.862436\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.48185i	0.136997i
$118$	0	0
$119$	23.6894	2.17161
$120$	0	0
$121$	22.9228	2.08389
$122$	0	0
$123$	− 4.24051i	− 0.382354i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.75678i	− 0.155889i	−0.996958	−	0.0779446i	$-0.975164\pi$
$127$	− 1.75678i	− 0.155889i	0.996958	−	0.0779446i	$-0.0248357\pi$
$128$	0	0
$129$	12.8605	1.13230
$130$	0	0
$131$	−6.14604	−0.536982	−0.268491	−	0.963282i	$-0.586525\pi$
$131$	−6.14604	−0.536982	−0.268491	+	0.963282i	$0.586525\pi$
$132$	0	0
$133$	6.99491i	0.606535i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.79739i	0.153561i	0.997048	+	0.0767806i	$0.0244641\pi$
$137$	1.79739i	0.153561i	−0.997048	+	0.0767806i	$0.975536\pi$
$138$	0	0
$139$	−15.6284	−1.32559	−0.662793	−	0.748803i	$-0.730629\pi$
$139$	−15.6284	−1.32559	−0.662793	+	0.748803i	$0.730629\pi$
$140$	0	0
$141$	1.59202	0.134073
$142$	0	0
$143$	31.1383i	2.60392i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 22.0193i	− 1.81612i
$148$	0	0
$149$	−20.3836	−1.66989	−0.834944	−	0.550335i	$-0.814500\pi$
$149$	−20.3836	−1.66989	−0.834944	+	0.550335i	$0.814500\pi$
$150$	0	0
$151$	−14.2100	−1.15639	−0.578196	−	0.815898i	$-0.696243\pi$
$151$	−14.2100	−1.15639	−0.578196	+	0.815898i	$0.696243\pi$
$152$	0	0
$153$	− 1.49994i	− 0.121263i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.01209i	− 0.479817i	−0.970796	−	0.239908i	$-0.922883\pi$
$157$	− 6.01209i	− 0.479817i	0.970796	−	0.239908i	$-0.0771174\pi$
$158$	0	0
$159$	−2.35776	−0.186982
$160$	0	0
$161$	−1.49215	−0.117598
$162$	0	0
$163$	12.5491i	0.982920i	0.870900	+	0.491460i	$0.163537\pi$
$163$	12.5491i	0.982920i	−0.870900	+	0.491460i	$0.836463\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.32243i	0.334480i	0.985916	+	0.167240i	$0.0534854\pi$
$167$	4.32243i	0.334480i	−0.985916	+	0.167240i	$0.946515\pi$
$168$	0	0
$169$	−15.5824	−1.19864
$170$	0	0
$171$	0.442894	0.0338689
$172$	0	0
$173$	8.22242i	0.625139i	0.949895	+	0.312570i	$0.101190\pi$
$173$	8.22242i	0.625139i	−0.949895	+	0.312570i	$0.898810\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 20.0447i	− 1.50666i
$178$	0	0
$179$	2.21704	0.165710	0.0828548	−	0.996562i	$-0.473596\pi$
$179$	2.21704	0.165710	0.0828548	+	0.996562i	$0.473596\pi$
$180$	0	0
$181$	15.6084	1.16016	0.580080	−	0.814559i	$-0.303021\pi$
$181$	15.6084	1.16016	0.580080	+	0.814559i	$0.303021\pi$
$182$	0	0
$183$	− 26.1449i	− 1.93268i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 31.5184i	− 2.30486i
$188$	0	0
$189$	21.5777	1.56955
$190$	0	0
$191$	0.823027	0.0595521	0.0297761	−	0.999557i	$-0.490521\pi$
$191$	0.823027	0.0595521	0.0297761	+	0.999557i	$0.490521\pi$
$192$	0	0
$193$	4.12278i	0.296764i	0.988930	+	0.148382i	$0.0474066\pi$
$193$	4.12278i	0.296764i	−0.988930	+	0.148382i	$0.952593\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.3961i	1.31067i	0.755340	+	0.655333i	$0.227472\pi$
$197$	18.3961i	1.31067i	−0.755340	+	0.655333i	$0.772528\pi$
$198$	0	0
$199$	1.01893	0.0722300	0.0361150	−	0.999348i	$-0.488502\pi$
$199$	1.01893	0.0722300	0.0361150	+	0.999348i	$0.488502\pi$
$200$	0	0
$201$	−5.45660	−0.384879
$202$	0	0
$203$	− 21.7316i	− 1.52526i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.0944776i	0.00656665i
$208$	0	0
$209$	9.30662	0.643752
$210$	0	0
$211$	8.14609	0.560800	0.280400	−	0.959883i	$-0.409533\pi$
$211$	8.14609	0.560800	0.280400	+	0.959883i	$0.409533\pi$
$212$	0	0
$213$	− 16.9683i	− 1.16265i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.3031i	− 1.37827i
$218$	0	0
$219$	−24.8294	−1.67782
$220$	0	0
$221$	28.9313	1.94613
$222$	0	0
$223$	− 12.4073i	− 0.830856i	−0.909626	−	0.415428i	$-0.863632\pi$
$223$	− 12.4073i	− 0.830856i	0.909626	−	0.415428i	$-0.136368\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.5589i	1.63003i	0.579438	+	0.815016i	$0.303272\pi$
$227$	24.5589i	1.63003i	−0.579438	+	0.815016i	$0.696728\pi$
$228$	0	0
$229$	5.35261	0.353711	0.176855	−	0.984237i	$-0.443408\pi$
$229$	5.35261	0.353711	0.176855	+	0.984237i	$0.443408\pi$
$230$	0	0
$231$	−46.1564	−3.03687
$232$	0	0
$233$	0.350300i	0.0229489i	0.999934	+	0.0114745i	$0.00365252\pi$
$233$	0.350300i	0.0229489i	−0.999934	+	0.0114745i	$0.996347\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.5920i	− 0.688025i
$238$	0	0
$239$	8.90513	0.576025	0.288012	−	0.957627i	$-0.407006\pi$
$239$	8.90513	0.576025	0.288012	+	0.957627i	$0.407006\pi$
$240$	0	0
$241$	−8.09465	−0.521422	−0.260711	−	0.965417i	$-0.583957\pi$
$241$	−8.09465	−0.521422	−0.260711	+	0.965417i	$0.583957\pi$
$242$	0	0
$243$	− 2.87154i	− 0.184209i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.54270i	0.543559i
$248$	0	0
$249$	−15.4701	−0.980378
$250$	0	0
$251$	−15.0480	−0.949821	−0.474910	−	0.880034i	$-0.657520\pi$
$251$	−15.0480	−0.949821	−0.474910	+	0.880034i	$0.657520\pi$
$252$	0	0
$253$	1.98528i	0.124813i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.1390i	− 0.819586i	−0.912179	−	0.409793i	$-0.865601\pi$
$257$	− 13.1390i	− 0.819586i	0.912179	−	0.409793i	$-0.134399\pi$
$258$	0	0
$259$	36.8146	2.28755
$260$	0	0
$261$	−1.37597	−0.0851706
$262$	0	0
$263$	16.3912i	1.01072i	0.862908	+	0.505362i	$0.168641\pi$
$263$	16.3912i	1.01072i	−0.862908	+	0.505362i	$0.831359\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	32.6265i	1.99671i
$268$	0	0
$269$	−1.56270	−0.0952793	−0.0476397	−	0.998865i	$-0.515170\pi$
$269$	−1.56270	−0.0952793	−0.0476397	+	0.998865i	$0.515170\pi$
$270$	0	0
$271$	−30.1625	−1.83224	−0.916121	−	0.400902i	$-0.868697\pi$
$271$	−30.1625	−1.83224	−0.916121	+	0.400902i	$0.868697\pi$
$272$	0	0
$273$	− 42.3677i	− 2.56421i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 17.9905i	− 1.08095i	−0.841361	−	0.540473i	$-0.818245\pi$
$277$	− 17.9905i	− 1.08095i	0.841361	−	0.540473i	$-0.181755\pi$
$278$	0	0
$279$	−1.28553	−0.0769624
$280$	0	0
$281$	−1.25760	−0.0750221	−0.0375111	−	0.999296i	$-0.511943\pi$
$281$	−1.25760	−0.0750221	−0.0375111	+	0.999296i	$0.511943\pi$
$282$	0	0
$283$	17.4528i	1.03746i	0.854938	+	0.518729i	$0.173595\pi$
$283$	17.4528i	1.03746i	−0.854938	+	0.518729i	$0.826405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.2543i	0.605290i
$288$	0	0
$289$	−12.2845	−0.722616
$290$	0	0
$291$	18.2694	1.07097
$292$	0	0
$293$	− 24.0216i	− 1.40336i	−0.712494	−	0.701678i	$-0.752435\pi$
$293$	− 24.0216i	− 1.40336i	0.712494	−	0.701678i	$-0.247565\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 28.7088i	− 1.66586i
$298$	0	0
$299$	−1.82232	−0.105387
$300$	0	0
$301$	−31.0988	−1.79250
$302$	0	0
$303$	− 12.0278i	− 0.690977i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.1584i	0.636845i	0.947949	+	0.318423i	$0.103153\pi$
$307$	11.1584i	0.636845i	−0.947949	+	0.318423i	$0.896847\pi$
$308$	0	0
$309$	17.8300	1.01432
$310$	0	0
$311$	−19.8088	−1.12326	−0.561628	−	0.827390i	$-0.689825\pi$
$311$	−19.8088	−1.12326	−0.561628	+	0.827390i	$0.689825\pi$
$312$	0	0
$313$	− 3.29732i	− 0.186376i	−0.995649	−	0.0931879i	$-0.970294\pi$
$313$	− 3.29732i	− 0.186376i	0.995649	−	0.0931879i	$-0.0297057\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.5911i	0.988017i	0.869457	+	0.494009i	$0.164469\pi$
$317$	17.5911i	0.988017i	−0.869457	+	0.494009i	$0.835531\pi$
$318$	0	0
$319$	−28.9136	−1.61885
$320$	0	0
$321$	−1.70423	−0.0951207
$322$	0	0
$323$	− 8.64698i	− 0.481131i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.5150i	− 1.02388i
$328$	0	0
$329$	−3.84978	−0.212245
$330$	0	0
$331$	−15.0330	−0.826288	−0.413144	−	0.910666i	$-0.635569\pi$
$331$	−15.0330	−0.826288	−0.413144	+	0.910666i	$0.635569\pi$
$332$	0	0
$333$	− 2.33098i	− 0.127737i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 31.7741i	− 1.73084i	−0.501043	−	0.865422i	$-0.667050\pi$
$337$	− 31.7741i	− 1.73084i	0.501043	−	0.865422i	$-0.332950\pi$
$338$	0	0
$339$	−16.1202	−0.875530
$340$	0	0
$341$	−27.0130	−1.46284
$342$	0	0
$343$	22.6032i	1.22046i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.6632i	− 1.43136i	−0.698430	−	0.715679i	$-0.746118\pi$
$347$	− 26.6632i	− 1.43136i	0.698430	−	0.715679i	$-0.253882\pi$
$348$	0	0
$349$	−4.44619	−0.237999	−0.119000	−	0.992894i	$-0.537969\pi$
$349$	−4.44619	−0.237999	−0.119000	+	0.992894i	$0.537969\pi$
$350$	0	0
$351$	26.3523	1.40658
$352$	0	0
$353$	− 0.316976i	− 0.0168710i	−0.999964	−	0.00843548i	$-0.997315\pi$
$353$	− 0.316976i	− 0.0168710i	0.999964	−	0.00843548i	$-0.00268513\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	42.8849i	2.26971i
$358$	0	0
$359$	22.0361	1.16302	0.581509	−	0.813540i	$-0.302462\pi$
$359$	22.0361	1.16302	0.581509	+	0.813540i	$0.302462\pi$
$360$	0	0
$361$	−16.4468	−0.865619
$362$	0	0
$363$	41.4971i	2.17803i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.44627i	0.232093i	0.993244	+	0.116047i	$0.0370222\pi$
$367$	4.44627i	0.232093i	−0.993244	+	0.116047i	$0.962978\pi$
$368$	0	0
$369$	0.649266	0.0337994
$370$	0	0
$371$	5.70145	0.296005
$372$	0	0
$373$	− 23.3308i	− 1.20802i	−0.796975	−	0.604012i	$-0.793568\pi$
$373$	− 23.3308i	− 1.20802i	0.796975	−	0.604012i	$-0.206432\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 26.5403i	− 1.36689i
$378$	0	0
$379$	19.0905	0.980614	0.490307	−	0.871550i	$-0.336885\pi$
$379$	19.0905	0.980614	0.490307	+	0.871550i	$0.336885\pi$
$380$	0	0
$381$	3.18030	0.162932
$382$	0	0
$383$	0.371944i	0.0190055i	0.999955	+	0.00950273i	$0.00302486\pi$
$383$	0.371944i	0.0190055i	−0.999955	+	0.00950273i	$0.996975\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.96907i	0.100093i
$388$	0	0
$389$	22.9338	1.16279	0.581395	−	0.813621i	$-0.302507\pi$
$389$	22.9338	1.16279	0.581395	+	0.813621i	$0.302507\pi$
$390$	0	0
$391$	1.84456	0.0932836
$392$	0	0
$393$	− 11.1262i	− 0.561240i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.2189i	0.713628i	0.934176	+	0.356814i	$0.116137\pi$
$397$	14.2189i	0.713628i	−0.934176	+	0.356814i	$0.883863\pi$
$398$	0	0
$399$	−12.6629	−0.633936
$400$	0	0
$401$	0.847710	0.0423326	0.0211663	−	0.999776i	$-0.493262\pi$
$401$	0.847710	0.0423326	0.0211663	+	0.999776i	$0.493262\pi$
$402$	0	0
$403$	− 24.7957i	− 1.23516i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 48.9813i	− 2.42791i
$408$	0	0
$409$	32.7324	1.61851	0.809256	−	0.587457i	$-0.199871\pi$
$409$	32.7324	1.61851	0.809256	+	0.587457i	$0.199871\pi$
$410$	0	0
$411$	−3.25381	−0.160498
$412$	0	0
$413$	48.4715i	2.38513i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 28.2921i	− 1.38547i
$418$	0	0
$419$	24.3482	1.18949	0.594743	−	0.803916i	$-0.297254\pi$
$419$	24.3482	1.18949	0.594743	+	0.803916i	$0.297254\pi$
$420$	0	0
$421$	−5.79282	−0.282325	−0.141162	−	0.989986i	$-0.545084\pi$
$421$	−5.79282	−0.282325	−0.141162	+	0.989986i	$0.545084\pi$
$422$	0	0
$423$	0.243755i	0.0118518i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	63.2226i	3.05956i
$428$	0	0
$429$	−56.3696	−2.72155
$430$	0	0
$431$	27.0917	1.30496	0.652480	−	0.757806i	$-0.273728\pi$
$431$	27.0917	1.30496	0.652480	+	0.757806i	$0.273728\pi$
$432$	0	0
$433$	35.6206i	1.71182i	0.517129	+	0.855908i	$0.327001\pi$
$433$	35.6206i	1.71182i	−0.517129	+	0.855908i	$0.672999\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.544654i	0.0260543i
$438$	0	0
$439$	−35.2574	−1.68275	−0.841373	−	0.540455i	$-0.818252\pi$
$439$	−35.2574	−1.68275	−0.841373	+	0.540455i	$0.818252\pi$
$440$	0	0
$441$	3.37138	0.160542
$442$	0	0
$443$	− 33.9602i	− 1.61350i	−0.590896	−	0.806748i	$-0.701225\pi$
$443$	− 33.9602i	− 1.61350i	0.590896	−	0.806748i	$-0.298775\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 36.9003i	− 1.74533i
$448$	0	0
$449$	−13.2446	−0.625050	−0.312525	−	0.949910i	$-0.601175\pi$
$449$	−13.2446	−0.625050	−0.312525	+	0.949910i	$0.601175\pi$
$450$	0	0
$451$	13.6432	0.642431
$452$	0	0
$453$	− 25.7243i	− 1.20863i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.86349i	− 0.133949i	−0.997755	−	0.0669743i	$-0.978665\pi$
$457$	− 2.86349i	− 0.133949i	0.997755	−	0.0669743i	$-0.0213345\pi$
$458$	0	0
$459$	−26.6740	−1.24504
$460$	0	0
$461$	22.8968	1.06641	0.533206	−	0.845986i	$-0.320987\pi$
$461$	22.8968	1.06641	0.533206	+	0.845986i	$0.320987\pi$
$462$	0	0
$463$	− 15.0334i	− 0.698659i	−0.937000	−	0.349330i	$-0.886409\pi$
$463$	− 15.0334i	− 0.698659i	0.937000	−	0.349330i	$-0.113591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.2490i	− 0.751913i	−0.926637	−	0.375957i	$-0.877314\pi$
$467$	− 16.2490i	− 0.751913i	0.926637	−	0.375957i	$-0.122686\pi$
$468$	0	0
$469$	13.1950	0.609287
$470$	0	0
$471$	10.8837	0.501493
$472$	0	0
$473$	41.3764i	1.90249i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.360997i	− 0.0165289i
$478$	0	0
$479$	16.8024	0.767722	0.383861	−	0.923391i	$-0.374594\pi$
$479$	16.8024	0.767722	0.383861	+	0.923391i	$0.374594\pi$
$480$	0	0
$481$	44.9607	2.05003
$482$	0	0
$483$	− 2.70123i	− 0.122910i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.0426i	0.772276i	0.922441	+	0.386138i	$0.126191\pi$
$487$	17.0426i	0.772276i	−0.922441	+	0.386138i	$0.873809\pi$
$488$	0	0
$489$	−22.7176	−1.02732
$490$	0	0
$491$	−25.5972	−1.15518	−0.577592	−	0.816326i	$-0.696007\pi$
$491$	−25.5972	−1.15518	−0.577592	+	0.816326i	$0.696007\pi$
$492$	0	0
$493$	26.8643i	1.20991i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	41.0321i	1.84054i
$498$	0	0
$499$	2.10330	0.0941565	0.0470783	−	0.998891i	$-0.485009\pi$
$499$	2.10330	0.0941565	0.0470783	+	0.998891i	$0.485009\pi$
$500$	0	0
$501$	−7.82489	−0.349590
$502$	0	0
$503$	− 1.80566i	− 0.0805105i	−0.999189	−	0.0402552i	$-0.987183\pi$
$503$	− 1.80566i	− 0.0805105i	0.999189	−	0.0402552i	$-0.0128171\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 28.2087i	− 1.25279i
$508$	0	0
$509$	16.5594	0.733980	0.366990	−	0.930225i	$-0.380388\pi$
$509$	16.5594	0.733980	0.366990	+	0.930225i	$0.380388\pi$
$510$	0	0
$511$	60.0417	2.65609
$512$	0	0
$513$	− 7.87618i	− 0.347742i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.12207i	0.225269i
$518$	0	0
$519$	−14.8850	−0.653380
$520$	0	0
$521$	−22.9830	−1.00690	−0.503452	−	0.864023i	$-0.667937\pi$
$521$	−22.9830	−1.00690	−0.503452	+	0.864023i	$0.667937\pi$
$522$	0	0
$523$	9.47128i	0.414150i	0.978325	+	0.207075i	$0.0663944\pi$
$523$	9.47128i	0.414150i	−0.978325	+	0.207075i	$0.933606\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.0984i	1.09330i
$528$	0	0
$529$	22.8838	0.994948
$530$	0	0
$531$	3.06905	0.133186
$532$	0	0
$533$	12.5233i	0.542443i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.01351i	0.173196i
$538$	0	0
$539$	70.8436	3.05145
$540$	0	0
$541$	13.2203	0.568386	0.284193	−	0.958767i	$-0.408274\pi$
$541$	13.2203	0.568386	0.284193	+	0.958767i	$0.408274\pi$
$542$	0	0
$543$	28.2558i	1.21257i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 33.1211i	− 1.41616i	−0.706133	−	0.708079i	$-0.749562\pi$
$547$	− 33.1211i	− 1.41616i	0.706133	−	0.708079i	$-0.250438\pi$
$548$	0	0
$549$	4.00304	0.170846
$550$	0	0
$551$	−7.93236	−0.337930
$552$	0	0
$553$	25.6133i	1.08919i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.75852i	0.371110i	0.982634	+	0.185555i	$0.0594083\pi$
$557$	8.75852i	0.371110i	−0.982634	+	0.185555i	$0.940592\pi$
$558$	0	0
$559$	−37.9801	−1.60639
$560$	0	0
$561$	57.0577	2.40898
$562$	0	0
$563$	1.05986i	0.0446678i	0.999751	+	0.0223339i	$0.00710970\pi$
$563$	1.05986i	0.0446678i	−0.999751	+	0.0223339i	$0.992890\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	42.7022i	1.79332i
$568$	0	0
$569$	−38.9002	−1.63078	−0.815390	−	0.578912i	$-0.803477\pi$
$569$	−38.9002	−1.63078	−0.815390	+	0.578912i	$0.803477\pi$
$570$	0	0
$571$	−33.4842	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$571$	−33.4842	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$572$	0	0
$573$	1.48992i	0.0622424i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.5217i	0.729436i	0.931118	+	0.364718i	$0.118835\pi$
$577$	17.5217i	0.729436i	−0.931118	+	0.364718i	$0.881165\pi$
$578$	0	0
$579$	−7.46346	−0.310171
$580$	0	0
$581$	37.4093	1.55200
$582$	0	0
$583$	− 7.58569i	− 0.314167i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21.6889i	− 0.895195i	−0.894235	−	0.447598i	$-0.852280\pi$
$587$	− 21.6889i	− 0.895195i	0.894235	−	0.447598i	$-0.147720\pi$
$588$	0	0
$589$	−7.41093	−0.305362
$590$	0	0
$591$	−33.3024	−1.36988
$592$	0	0
$593$	− 34.8683i	− 1.43187i	−0.698167	−	0.715935i	$-0.746001\pi$
$593$	− 34.8683i	− 1.43187i	0.698167	−	0.715935i	$-0.253999\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.84456i	0.0754930i
$598$	0	0
$599$	14.6435	0.598317	0.299158	−	0.954203i	$-0.403294\pi$
$599$	14.6435	0.598317	0.299158	+	0.954203i	$0.403294\pi$
$600$	0	0
$601$	−9.69821	−0.395598	−0.197799	−	0.980243i	$-0.563379\pi$
$601$	−9.69821	−0.395598	−0.197799	+	0.980243i	$0.563379\pi$
$602$	0	0
$603$	− 0.835460i	− 0.0340226i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.94086i	0.119366i	0.998217	+	0.0596829i	$0.0190090\pi$
$607$	2.94086i	0.119366i	−0.998217	+	0.0596829i	$0.980991\pi$
$608$	0	0
$609$	39.3407	1.59417
$610$	0	0
$611$	−4.70163	−0.190208
$612$	0	0
$613$	17.0019i	0.686702i	0.939207	+	0.343351i	$0.111562\pi$
$613$	17.0019i	0.686702i	−0.939207	+	0.343351i	$0.888438\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.7996i	0.716583i	0.933610	+	0.358292i	$0.116641\pi$
$617$	17.7996i	0.716583i	−0.933610	+	0.358292i	$0.883359\pi$
$618$	0	0
$619$	27.5237	1.10627	0.553135	−	0.833092i	$-0.313431\pi$
$619$	27.5237	1.10627	0.553135	+	0.833092i	$0.313431\pi$
$620$	0	0
$621$	1.68014	0.0674215
$622$	0	0
$623$	− 78.8962i	− 3.16091i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.8477i	0.672834i
$628$	0	0
$629$	−45.5096	−1.81458
$630$	0	0
$631$	47.3515	1.88503	0.942517	−	0.334158i	$-0.108452\pi$
$631$	47.3515	1.88503	0.942517	+	0.334158i	$0.108452\pi$
$632$	0	0
$633$	14.7468i	0.586134i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	65.0285i	2.57652i
$638$	0	0
$639$	2.59801	0.102776
$640$	0	0
$641$	−15.3183	−0.605037	−0.302518	−	0.953144i	$-0.597827\pi$
$641$	−15.3183	−0.605037	−0.302518	+	0.953144i	$0.597827\pi$
$642$	0	0
$643$	38.4573i	1.51661i	0.651901	+	0.758304i	$0.273972\pi$
$643$	38.4573i	1.51661i	−0.651901	+	0.758304i	$0.726028\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.46190i	0.136101i	0.997682	+	0.0680507i	$0.0216780\pi$
$647$	3.46190i	0.136101i	−0.997682	+	0.0680507i	$0.978322\pi$
$648$	0	0
$649$	64.4906	2.53148
$650$	0	0
$651$	36.7547	1.44053
$652$	0	0
$653$	37.9039i	1.48329i	0.670790	+	0.741647i	$0.265955\pi$
$653$	37.9039i	1.48329i	−0.670790	+	0.741647i	$0.734045\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.80164i	− 0.148316i
$658$	0	0
$659$	−22.2876	−0.868203	−0.434102	−	0.900864i	$-0.642934\pi$
$659$	−22.2876	−0.868203	−0.434102	+	0.900864i	$0.642934\pi$
$660$	0	0
$661$	−10.1877	−0.396257	−0.198129	−	0.980176i	$-0.563486\pi$
$661$	−10.1877	−0.396257	−0.198129	+	0.980176i	$0.563486\pi$
$662$	0	0
$663$	52.3742i	2.03405i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.69212i	− 0.0655192i
$668$	0	0
$669$	22.4609	0.868390
$670$	0	0
$671$	84.1168	3.24729
$672$	0	0
$673$	− 22.6737i	− 0.874008i	−0.899460	−	0.437004i	$-0.856040\pi$
$673$	− 22.6737i	− 0.874008i	0.899460	−	0.437004i	$-0.143960\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.30011i	− 0.318999i	−0.987198	−	0.159500i	$-0.949012\pi$
$677$	− 8.30011i	− 0.318999i	0.987198	−	0.159500i	$-0.0509881\pi$
$678$	0	0
$679$	−44.1784	−1.69541
$680$	0	0
$681$	−44.4589	−1.70367
$682$	0	0
$683$	8.62826i	0.330151i	0.986281	+	0.165076i	$0.0527868\pi$
$683$	8.62826i	0.330151i	−0.986281	+	0.165076i	$0.947213\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.68982i	0.369690i
$688$	0	0
$689$	6.96303	0.265270
$690$	0	0
$691$	42.6346	1.62190	0.810949	−	0.585117i	$-0.198951\pi$
$691$	42.6346	1.62190	0.810949	+	0.585117i	$0.198951\pi$
$692$	0	0
$693$	− 7.06701i	− 0.268454i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12.6762i	− 0.480143i
$698$	0	0
$699$	−0.634147	−0.0239857
$700$	0	0
$701$	−38.4312	−1.45153	−0.725763	−	0.687945i	$-0.758513\pi$
$701$	−38.4312	−1.45153	−0.725763	+	0.687945i	$0.758513\pi$
$702$	0	0
$703$	− 13.4379i	− 0.506818i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.0852i	1.09386i
$708$	0	0
$709$	−19.9047	−0.747537	−0.373769	−	0.927522i	$-0.621935\pi$
$709$	−19.9047	−0.747537	−0.373769	+	0.927522i	$0.621935\pi$
$710$	0	0
$711$	1.62175	0.0608202
$712$	0	0
$713$	− 1.58089i	− 0.0592048i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.1209i	0.602047i
$718$	0	0
$719$	14.9260	0.556644	0.278322	−	0.960488i	$-0.410222\pi$
$719$	14.9260	0.556644	0.278322	+	0.960488i	$0.410222\pi$
$720$	0	0
$721$	−43.1160	−1.60572
$722$	0	0
$723$	− 14.6537i	− 0.544978i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.6583i	0.951615i	0.879549	+	0.475808i	$0.157844\pi$
$727$	25.6583i	0.951615i	−0.879549	+	0.475808i	$0.842156\pi$
$728$	0	0
$729$	−24.0658	−0.891325
$730$	0	0
$731$	38.4437	1.42189
$732$	0	0
$733$	− 45.1247i	− 1.66672i	−0.552731	−	0.833360i	$-0.686414\pi$
$733$	− 45.1247i	− 1.66672i	0.552731	−	0.833360i	$-0.313586\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 17.5557i	− 0.646672i
$738$	0	0
$739$	−4.88625	−0.179743	−0.0898717	−	0.995953i	$-0.528646\pi$
$739$	−4.88625	−0.179743	−0.0898717	+	0.995953i	$0.528646\pi$
$740$	0	0
$741$	−15.4648	−0.568114
$742$	0	0
$743$	31.3357i	1.14960i	0.818295	+	0.574798i	$0.194919\pi$
$743$	31.3357i	1.14960i	−0.818295	+	0.574798i	$0.805081\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.36863i	− 0.0866637i
$748$	0	0
$749$	4.12111	0.150582
$750$	0	0
$751$	−2.75676	−0.100596	−0.0502978	−	0.998734i	$-0.516017\pi$
$751$	−2.75676	−0.100596	−0.0502978	+	0.998734i	$0.516017\pi$
$752$	0	0
$753$	− 27.2413i	− 0.992729i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.7888i	1.01000i	0.863119	+	0.505000i	$0.168508\pi$
$757$	27.7888i	1.01000i	−0.863119	+	0.505000i	$0.831492\pi$
$758$	0	0
$759$	−3.59394	−0.130452
$760$	0	0
$761$	38.2357	1.38604	0.693022	−	0.720916i	$-0.256279\pi$
$761$	38.2357	1.38604	0.693022	+	0.720916i	$0.256279\pi$
$762$	0	0
$763$	44.7724i	1.62087i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	59.1970i	2.13748i
$768$	0	0
$769$	−41.3417	−1.49082	−0.745409	−	0.666607i	$-0.767746\pi$
$769$	−41.3417	−1.49082	−0.745409	+	0.666607i	$0.767746\pi$
$770$	0	0
$771$	23.7854	0.856611
$772$	0	0
$773$	− 4.31886i	− 0.155339i	−0.996979	−	0.0776693i	$-0.975252\pi$
$773$	− 4.31886i	− 0.155339i	0.996979	−	0.0776693i	$-0.0247478\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	66.6453i	2.39089i
$778$	0	0
$779$	3.74296	0.134105
$780$	0	0
$781$	54.5926	1.95348
$782$	0	0
$783$	24.4695i	0.874470i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.48163i	− 0.302337i	−0.988508	−	0.151169i	$-0.951696\pi$
$787$	− 8.48163i	− 0.302337i	0.988508	−	0.151169i	$-0.0483037\pi$
$788$	0	0
$789$	−29.6729	−1.05638
$790$	0	0
$791$	38.9814	1.38602
$792$	0	0
$793$	77.2121i	2.74188i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13.6851i	− 0.484750i	−0.970183	−	0.242375i	$-0.922074\pi$
$797$	− 13.6851i	− 0.484750i	0.970183	−	0.242375i	$-0.0779265\pi$
$798$	0	0
$799$	4.75903	0.168362
$800$	0	0
$801$	−4.99544	−0.176505
$802$	0	0
$803$	− 79.8846i	− 2.81907i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.82895i	− 0.0995836i
$808$	0	0
$809$	5.08132	0.178650	0.0893249	−	0.996003i	$-0.471529\pi$
$809$	5.08132	0.178650	0.0893249	+	0.996003i	$0.471529\pi$
$810$	0	0
$811$	−24.9990	−0.877834	−0.438917	−	0.898528i	$-0.644638\pi$
$811$	−24.9990	−0.877834	−0.438917	+	0.898528i	$0.644638\pi$
$812$	0	0
$813$	− 54.6031i	− 1.91501i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.3515i	0.397138i
$818$	0	0
$819$	6.48693	0.226672
$820$	0	0
$821$	−25.8738	−0.903002	−0.451501	−	0.892271i	$-0.649111\pi$
$821$	−25.8738	−0.903002	−0.451501	+	0.892271i	$0.649111\pi$
$822$	0	0
$823$	11.0152i	0.383965i	0.981398	+	0.191982i	$0.0614916\pi$
$823$	11.0152i	0.383965i	−0.981398	+	0.191982i	$0.938508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.5502i	− 0.471185i	−0.971852	−	0.235592i	$-0.924297\pi$
$827$	− 13.5502i	− 0.471185i	0.971852	−	0.235592i	$-0.0757031\pi$
$828$	0	0
$829$	38.5707	1.33962	0.669808	−	0.742534i	$-0.266376\pi$
$829$	38.5707	1.33962	0.669808	+	0.742534i	$0.266376\pi$
$830$	0	0
$831$	32.5682	1.12978
$832$	0	0
$833$	− 65.8223i	− 2.28061i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	22.8611i	0.790194i
$838$	0	0
$839$	−4.38927	−0.151535	−0.0757673	−	0.997126i	$-0.524141\pi$
$839$	−4.38927	−0.151535	−0.0757673	+	0.997126i	$0.524141\pi$
$840$	0	0
$841$	−4.35591	−0.150204
$842$	0	0
$843$	− 2.27663i	− 0.0784113i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 100.347i	− 3.44796i
$848$	0	0
$849$	−31.5947	−1.08433
$850$	0	0
$851$	2.86655	0.0982639
$852$	0	0
$853$	− 35.3589i	− 1.21067i	−0.795972	−	0.605333i	$-0.793040\pi$
$853$	− 35.3589i	− 1.21067i	0.795972	−	0.605333i	$-0.206960\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.4133i	− 1.92704i	−0.267636	−	0.963520i	$-0.586242\pi$
$857$	− 56.4133i	− 1.92704i	0.267636	−	0.963520i	$-0.413758\pi$
$858$	0	0
$859$	28.2629	0.964319	0.482160	−	0.876083i	$-0.339853\pi$
$859$	28.2629	0.964319	0.482160	+	0.876083i	$0.339853\pi$
$860$	0	0
$861$	−18.5633	−0.632635
$862$	0	0
$863$	11.3915i	0.387772i	0.981024	+	0.193886i	$0.0621092\pi$
$863$	11.3915i	0.387772i	−0.981024	+	0.193886i	$0.937891\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 22.2385i	− 0.755261i
$868$	0	0
$869$	34.0781	1.15602
$870$	0	0
$871$	16.1147	0.546024
$872$	0	0
$873$	2.79723i	0.0946718i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.1016i	0.577479i	0.957408	+	0.288740i	$0.0932362\pi$
$877$	17.1016i	0.577479i	−0.957408	+	0.288740i	$0.906764\pi$
$878$	0	0
$879$	43.4862	1.46675
$880$	0	0
$881$	−53.3569	−1.79764	−0.898821	−	0.438317i	$-0.855575\pi$
$881$	−53.3569	−1.79764	−0.898821	+	0.438317i	$0.855575\pi$
$882$	0	0
$883$	− 35.9395i	− 1.20946i	−0.796430	−	0.604730i	$-0.793281\pi$
$883$	− 35.9395i	− 1.20946i	0.796430	−	0.604730i	$-0.206719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 48.4603i	− 1.62714i	−0.581468	−	0.813569i	$-0.697521\pi$
$887$	− 48.4603i	− 1.62714i	0.581468	−	0.813569i	$-0.302479\pi$
$888$	0	0
$889$	−7.69049	−0.257931
$890$	0	0
$891$	56.8146	1.90336
$892$	0	0
$893$	1.40522i	0.0470241i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.29894i	− 0.110148i
$898$	0	0
$899$	23.0241	0.767898
$900$	0	0
$901$	−7.04803	−0.234804
$902$	0	0
$903$	− 56.2980i	− 1.87348i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21.2305i	− 0.704948i	−0.935822	−	0.352474i	$-0.885341\pi$
$907$	− 21.2305i	− 0.704948i	0.935822	−	0.352474i	$-0.114659\pi$
$908$	0	0
$909$	1.84157	0.0610811
$910$	0	0
$911$	−23.2599	−0.770634	−0.385317	−	0.922784i	$-0.625908\pi$
$911$	−23.2599	−0.770634	−0.385317	+	0.922784i	$0.625908\pi$
$912$	0	0
$913$	− 49.7725i	− 1.64723i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.9049i	0.888478i
$918$	0	0
$919$	−43.5439	−1.43638	−0.718191	−	0.695846i	$-0.755029\pi$
$919$	−43.5439	−1.43638	−0.718191	+	0.695846i	$0.755029\pi$
$920$	0	0
$921$	−20.2001	−0.665615
$922$	0	0
$923$	50.1114i	1.64944i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.72996i	0.0896637i
$928$	0	0
$929$	−1.93781	−0.0635776	−0.0317888	−	0.999495i	$-0.510120\pi$
$929$	−1.93781	−0.0635776	−0.0317888	+	0.999495i	$0.510120\pi$
$930$	0	0
$931$	19.4357	0.636980
$932$	0	0
$933$	− 35.8599i	− 1.17400i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.2146i	0.399032i	0.979895	+	0.199516i	$0.0639370\pi$
$937$	12.2146i	0.399032i	−0.979895	+	0.199516i	$0.936063\pi$
$938$	0	0
$939$	5.96913	0.194795
$940$	0	0
$941$	19.4761	0.634902	0.317451	−	0.948275i	$-0.397173\pi$
$941$	19.4761	0.634902	0.317451	+	0.948275i	$0.397173\pi$
$942$	0	0
$943$	0.798442i	0.0260009i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.77050i	0.0575335i	0.999586	+	0.0287667i	$0.00915800\pi$
$947$	1.77050i	0.0575335i	−0.999586	+	0.0287667i	$0.990842\pi$
$948$	0	0
$949$	73.3274	2.38031
$950$	0	0
$951$	−31.8452	−1.03265
$952$	0	0
$953$	− 26.4519i	− 0.856860i	−0.903575	−	0.428430i	$-0.859067\pi$
$953$	− 26.4519i	− 0.856860i	0.903575	−	0.428430i	$-0.140933\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 52.3422i	− 1.69198i
$958$	0	0
$959$	7.86824	0.254079
$960$	0	0
$961$	−9.48935	−0.306108
$962$	0	0
$963$	− 0.260935i	− 0.00840850i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.5985i	0.951822i	0.879493	+	0.475911i	$0.157882\pi$
$967$	29.5985i	0.951822i	−0.879493	+	0.475911i	$0.842118\pi$
$968$	0	0
$969$	15.6536	0.502866
$970$	0	0
$971$	−3.26775	−0.104867	−0.0524335	−	0.998624i	$-0.516698\pi$
$971$	−3.26775	−0.104867	−0.0524335	+	0.998624i	$0.516698\pi$
$972$	0	0
$973$	68.4150i	2.19328i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11.6591i	− 0.373009i	−0.982454	−	0.186504i	$-0.940284\pi$
$977$	− 11.6591i	− 0.373009i	0.982454	−	0.186504i	$-0.0597158\pi$
$978$	0	0
$979$	−104.970	−3.35486
$980$	0	0
$981$	2.83483	0.0905093
$982$	0	0
$983$	− 37.8109i	− 1.20598i	−0.797749	−	0.602990i	$-0.793976\pi$
$983$	− 37.8109i	− 1.20598i	0.797749	−	0.602990i	$-0.206024\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.96925i	− 0.221834i
$988$	0	0
$989$	−2.42149	−0.0769988
$990$	0	0
$991$	10.0303	0.318623	0.159311	−	0.987228i	$-0.449073\pi$
$991$	10.0303	0.318623	0.159311	+	0.987228i	$0.449073\pi$
$992$	0	0
$993$	− 27.2142i	− 0.863616i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 45.2025i	− 1.43158i	−0.698317	−	0.715788i	$-0.746067\pi$
$997$	− 45.2025i	− 1.43158i	0.698317	−	0.715788i	$-0.253933\pi$
$998$	0	0
$999$	−41.4528	−1.31151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.c.f.1249.9		12
4.3	odd	2		4000.2.c.g.1249.4		12
5.2	odd	4		4000.2.a.m.1.5	yes	6
5.3	odd	4		4000.2.a.k.1.2	✓	6
5.4	even	2	inner	4000.2.c.f.1249.4		12
20.3	even	4		4000.2.a.n.1.5	yes	6
20.7	even	4		4000.2.a.l.1.2	yes	6
20.19	odd	2		4000.2.c.g.1249.9		12
40.3	even	4		8000.2.a.bw.1.2		6
40.13	odd	4		8000.2.a.bv.1.5		6
40.27	even	4		8000.2.a.bu.1.5		6
40.37	odd	4		8000.2.a.bx.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.k.1.2	✓	6	5.3	odd	4
4000.2.a.l.1.2	yes	6	20.7	even	4
4000.2.a.m.1.5	yes	6	5.2	odd	4
4000.2.a.n.1.5	yes	6	20.3	even	4
4000.2.c.f.1249.4		12	5.4	even	2	inner
4000.2.c.f.1249.9		12	1.1	even	1	trivial
4000.2.c.g.1249.4		12	4.3	odd	2
4000.2.c.g.1249.9		12	20.19	odd	2
8000.2.a.bu.1.5		6	40.27	even	4
8000.2.a.bv.1.5		6	40.13	odd	4
8000.2.a.bw.1.2		6	40.3	even	4
8000.2.a.bx.1.2		6	40.37	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.81030i q^{3} -4.37760i q^{7} -0.277175 q^{9} -5.82433 q^{11} -5.34625i q^{13} +5.41151i q^{17} -1.59789 q^{19} +7.92476 q^{21} -0.340859i q^{23} +4.92912i q^{27} +4.96428 q^{29} +4.63796 q^{31} -10.5438i q^{33} +8.40977i q^{37} +9.67829 q^{39} -2.34244 q^{41} -7.10407i q^{43} -0.879427i q^{47} -12.1634 q^{49} -9.79645 q^{51} +1.30241i q^{53} -2.89265i q^{57} -11.0726 q^{59} -14.4423 q^{61} +1.21336i q^{63} +3.01420i q^{67} +0.617056 q^{69} -9.37319 q^{71} +13.7157i q^{73} +25.4966i q^{77} -5.85098 q^{79} -9.75470 q^{81} +8.54562i q^{83} +8.98682i q^{87} +18.0227 q^{89} -23.4037 q^{91} +8.39608i q^{93} -10.0919i q^{97} +1.61436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9} - 26 q^{11} + 18 q^{19} - 4 q^{29} - 24 q^{31} + 52 q^{39} - 10 q^{41} + 6 q^{49} - 36 q^{51} + 50 q^{59} - 12 q^{61} + 24 q^{69} - 68 q^{71} + 32 q^{79} - 36 q^{81} + 6 q^{89} - 52 q^{91}+ \cdots + 86 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 - 26 * q^11 + 18 * q^19 - 4 * q^29 - 24 * q^31 + 52 * q^39 - 10 * q^41 + 6 * q^49 - 36 * q^51 + 50 * q^59 - 12 * q^61 + 24 * q^69 - 68 * q^71 + 32 * q^79 - 36 * q^81 + 6 * q^89 - 52 * q^91 + 86 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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