LMFDB - Embedded newform 6000.2.f.h.1249.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6000,2,Mod(1249,6000)] 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("6000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,2,0,0,0,0,0,0,0,10,0,2] 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:	$47.9102412128$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 750)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.h.1249.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.618034i q^{7} -1.00000 q^{9} +1.61803 q^{11} +4.85410i q^{13} -0.763932i q^{17} +5.85410 q^{19} -0.618034 q^{21} -4.85410i q^{23} +1.00000i q^{27} +2.76393 q^{29} +2.47214 q^{31} -1.61803i q^{33} +9.56231i q^{37} +4.85410 q^{39} -9.38197 q^{41} -5.70820i q^{43} +1.61803i q^{47} +6.61803 q^{49} -0.763932 q^{51} +5.38197i q^{53} -5.85410i q^{57} -13.0902 q^{59} -9.70820 q^{61} +0.618034i q^{63} -3.70820i q^{67} -4.85410 q^{69} +3.52786 q^{71} +12.9443i q^{73} -1.00000i q^{77} +13.4164 q^{79} +1.00000 q^{81} +14.9443i q^{83} -2.76393i q^{87} +18.0902 q^{89} +3.00000 q^{91} -2.47214i q^{93} -9.70820i q^{97} -1.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 2 q^{11} + 10 q^{19} + 2 q^{21} + 20 q^{29} - 8 q^{31} + 6 q^{39} - 42 q^{41} + 22 q^{49} - 12 q^{51} - 30 q^{59} - 12 q^{61} - 6 q^{69} + 32 q^{71} + 4 q^{81} + 50 q^{89} + 12 q^{91} - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 2 * q^11 + 10 * q^19 + 2 * q^21 + 20 * q^29 - 8 * q^31 + 6 * q^39 - 42 * q^41 + 22 * q^49 - 12 * q^51 - 30 * q^59 - 12 * q^61 - 6 * q^69 + 32 * q^71 + 4 * q^81 + 50 * q^89 + 12 * q^91 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times$.

$n$	$751$	$4001$	$4501$	$5377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.618034i	− 0.233595i	−0.993156	−	0.116797i	$-0.962737\pi$
$7$	− 0.618034i	− 0.233595i	0.993156	−	0.116797i	$-0.0372628\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.61803	0.487856	0.243928	−	0.969793i	$-0.421564\pi$
$11$	1.61803	0.487856	0.243928	+	0.969793i	$0.421564\pi$
$12$	0	0
$13$	4.85410i	1.34629i	0.739512	+	0.673143i	$0.235056\pi$
$13$	4.85410i	1.34629i	−0.739512	+	0.673143i	$0.764944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.763932i	− 0.185281i	−0.995700	−	0.0926404i	$-0.970469\pi$
$17$	− 0.763932i	− 0.185281i	0.995700	−	0.0926404i	$-0.0295307\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−0.618034	−0.134866
$22$	0	0
$23$	− 4.85410i	− 1.01215i	−0.862489	−	0.506075i	$-0.831096\pi$
$23$	− 4.85410i	− 1.01215i	0.862489	−	0.506075i	$-0.168904\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	0	0
$33$	− 1.61803i	− 0.281664i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.56231i	1.57203i	0.618205	+	0.786017i	$0.287860\pi$
$37$	9.56231i	1.57203i	−0.618205	+	0.786017i	$0.712140\pi$
$38$	0	0
$39$	4.85410	0.777278
$40$	0	0
$41$	−9.38197	−1.46522	−0.732608	−	0.680650i	$-0.761697\pi$
$41$	−9.38197	−1.46522	−0.732608	+	0.680650i	$0.761697\pi$
$42$	0	0
$43$	− 5.70820i	− 0.870493i	−0.900311	−	0.435246i	$-0.856661\pi$
$43$	− 5.70820i	− 0.870493i	0.900311	−	0.435246i	$-0.143339\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.61803i	0.236015i	0.993013	+	0.118007i	$0.0376506\pi$
$47$	1.61803i	0.236015i	−0.993013	+	0.118007i	$0.962349\pi$
$48$	0	0
$49$	6.61803	0.945433
$50$	0	0
$51$	−0.763932	−0.106972
$52$	0	0
$53$	5.38197i	0.739270i	0.929177	+	0.369635i	$0.120517\pi$
$53$	5.38197i	0.739270i	−0.929177	+	0.369635i	$0.879483\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.85410i	− 0.775395i
$58$	0	0
$59$	−13.0902	−1.70419	−0.852097	−	0.523383i	$-0.824670\pi$
$59$	−13.0902	−1.70419	−0.852097	+	0.523383i	$0.824670\pi$
$60$	0	0
$61$	−9.70820	−1.24301	−0.621504	−	0.783411i	$-0.713478\pi$
$61$	−9.70820	−1.24301	−0.621504	+	0.783411i	$0.713478\pi$
$62$	0	0
$63$	0.618034i	0.0778650i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.70820i	− 0.453029i	−0.974008	−	0.226515i	$-0.927267\pi$
$67$	− 3.70820i	− 0.453029i	0.974008	−	0.226515i	$-0.0727331\pi$
$68$	0	0
$69$	−4.85410	−0.584365
$70$	0	0
$71$	3.52786	0.418680	0.209340	−	0.977843i	$-0.432868\pi$
$71$	3.52786	0.418680	0.209340	+	0.977843i	$0.432868\pi$
$72$	0	0
$73$	12.9443i	1.51501i	0.652828	+	0.757506i	$0.273582\pi$
$73$	12.9443i	1.51501i	−0.652828	+	0.757506i	$0.726418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.00000i	− 0.113961i
$78$	0	0
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.9443i	1.64035i	0.572115	+	0.820173i	$0.306123\pi$
$83$	14.9443i	1.64035i	−0.572115	+	0.820173i	$0.693877\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.76393i	− 0.296325i
$88$	0	0
$89$	18.0902	1.91755	0.958777	−	0.284159i	$-0.0917144\pi$
$89$	18.0902	1.91755	0.958777	+	0.284159i	$0.0917144\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	− 2.47214i	− 0.256349i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 9.70820i	− 0.985719i	−0.870109	−	0.492859i	$-0.835952\pi$
$97$	− 9.70820i	− 0.985719i	0.870109	−	0.492859i	$-0.164048\pi$
$98$	0	0
$99$	−1.61803	−0.162619
$100$	0	0
$101$	12.6525	1.25897	0.629484	−	0.777013i	$-0.283266\pi$
$101$	12.6525	1.25897	0.629484	+	0.777013i	$0.283266\pi$
$102$	0	0
$103$	8.56231i	0.843669i	0.906673	+	0.421835i	$0.138614\pi$
$103$	8.56231i	0.843669i	−0.906673	+	0.421835i	$0.861386\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.94427i	0.671328i	0.941982	+	0.335664i	$0.108961\pi$
$107$	6.94427i	0.671328i	−0.941982	+	0.335664i	$0.891039\pi$
$108$	0	0
$109$	6.18034	0.591969	0.295985	−	0.955193i	$-0.404352\pi$
$109$	6.18034	0.591969	0.295985	+	0.955193i	$0.404352\pi$
$110$	0	0
$111$	9.56231	0.907614
$112$	0	0
$113$	− 3.23607i	− 0.304424i	−0.988348	−	0.152212i	$-0.951360\pi$
$113$	− 3.23607i	− 0.304424i	0.988348	−	0.152212i	$-0.0486396\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.85410i	− 0.448762i
$118$	0	0
$119$	−0.472136	−0.0432806
$120$	0	0
$121$	−8.38197	−0.761997
$122$	0	0
$123$	9.38197i	0.845943i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.4164i	1.01304i	0.862228	+	0.506521i	$0.169069\pi$
$127$	11.4164i	1.01304i	−0.862228	+	0.506521i	$0.830931\pi$
$128$	0	0
$129$	−5.70820	−0.502579
$130$	0	0
$131$	−3.90983	−0.341603	−0.170802	−	0.985305i	$-0.554636\pi$
$131$	−3.90983	−0.341603	−0.170802	+	0.985305i	$0.554636\pi$
$132$	0	0
$133$	− 3.61803i	− 0.313723i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.05573i	0.261068i	0.991444	+	0.130534i	$0.0416692\pi$
$137$	3.05573i	0.261068i	−0.991444	+	0.130534i	$0.958331\pi$
$138$	0	0
$139$	12.5623	1.06552	0.532760	−	0.846266i	$-0.321155\pi$
$139$	12.5623	1.06552	0.532760	+	0.846266i	$0.321155\pi$
$140$	0	0
$141$	1.61803	0.136263
$142$	0	0
$143$	7.85410i	0.656793i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.61803i	− 0.545846i
$148$	0	0
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	−10.2918	−0.837534	−0.418767	−	0.908094i	$-0.637538\pi$
$151$	−10.2918	−0.837534	−0.418767	+	0.908094i	$0.637538\pi$
$152$	0	0
$153$	0.763932i	0.0617602i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.9443i	0.873448i	0.899596	+	0.436724i	$0.143861\pi$
$157$	10.9443i	0.873448i	−0.899596	+	0.436724i	$0.856139\pi$
$158$	0	0
$159$	5.38197	0.426818
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	4.94427i	0.387265i	0.981074	+	0.193633i	$0.0620270\pi$
$163$	4.94427i	0.387265i	−0.981074	+	0.193633i	$0.937973\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.85410i	− 0.607769i	−0.952709	−	0.303884i	$-0.901716\pi$
$167$	− 7.85410i	− 0.607769i	0.952709	−	0.303884i	$-0.0982836\pi$
$168$	0	0
$169$	−10.5623	−0.812485
$170$	0	0
$171$	−5.85410	−0.447674
$172$	0	0
$173$	− 14.6180i	− 1.11139i	−0.831387	−	0.555694i	$-0.812453\pi$
$173$	− 14.6180i	− 1.11139i	0.831387	−	0.555694i	$-0.187547\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.0902i	0.983917i
$178$	0	0
$179$	22.0344	1.64693	0.823466	−	0.567366i	$-0.192038\pi$
$179$	22.0344	1.64693	0.823466	+	0.567366i	$0.192038\pi$
$180$	0	0
$181$	25.4164	1.88919	0.944593	−	0.328243i	$-0.106456\pi$
$181$	25.4164	1.88919	0.944593	+	0.328243i	$0.106456\pi$
$182$	0	0
$183$	9.70820i	0.717651i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.23607i	− 0.0903902i
$188$	0	0
$189$	0.618034	0.0449554
$190$	0	0
$191$	16.9443	1.22604	0.613022	−	0.790066i	$-0.289954\pi$
$191$	16.9443	1.22604	0.613022	+	0.790066i	$0.289954\pi$
$192$	0	0
$193$	− 4.29180i	− 0.308930i	−0.987998	−	0.154465i	$-0.950635\pi$
$193$	− 4.29180i	− 0.308930i	0.987998	−	0.154465i	$-0.0493654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.94427i	− 0.494759i	−0.968919	−	0.247379i	$-0.920431\pi$
$197$	− 6.94427i	− 0.494759i	0.968919	−	0.247379i	$-0.0795694\pi$
$198$	0	0
$199$	−6.18034	−0.438113	−0.219056	−	0.975712i	$-0.570298\pi$
$199$	−6.18034	−0.438113	−0.219056	+	0.975712i	$0.570298\pi$
$200$	0	0
$201$	−3.70820	−0.261557
$202$	0	0
$203$	− 1.70820i	− 0.119892i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.85410i	0.337383i
$208$	0	0
$209$	9.47214	0.655201
$210$	0	0
$211$	17.2705	1.18895	0.594475	−	0.804114i	$-0.297360\pi$
$211$	17.2705	1.18895	0.594475	+	0.804114i	$0.297360\pi$
$212$	0	0
$213$	− 3.52786i	− 0.241725i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.52786i	− 0.103718i
$218$	0	0
$219$	12.9443	0.874693
$220$	0	0
$221$	3.70820	0.249441
$222$	0	0
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.1803i	− 1.87039i	−0.354127	−	0.935197i	$-0.615222\pi$
$227$	− 28.1803i	− 1.87039i	0.354127	−	0.935197i	$-0.384778\pi$
$228$	0	0
$229$	−1.70820	−0.112881	−0.0564406	−	0.998406i	$-0.517975\pi$
$229$	−1.70820	−0.112881	−0.0564406	+	0.998406i	$0.517975\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	− 19.4164i	− 1.27201i	−0.771684	−	0.636006i	$-0.780586\pi$
$233$	− 19.4164i	− 1.27201i	0.771684	−	0.636006i	$-0.219414\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 13.4164i	− 0.871489i
$238$	0	0
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	−7.14590	−0.460308	−0.230154	−	0.973154i	$-0.573923\pi$
$241$	−7.14590	−0.460308	−0.230154	+	0.973154i	$0.573923\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.4164i	1.80809i
$248$	0	0
$249$	14.9443	0.947055
$250$	0	0
$251$	−20.9443	−1.32199	−0.660995	−	0.750390i	$-0.729866\pi$
$251$	−20.9443	−1.32199	−0.660995	+	0.750390i	$0.729866\pi$
$252$	0	0
$253$	− 7.85410i	− 0.493783i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.5279i	− 0.843845i	−0.906632	−	0.421922i	$-0.861355\pi$
$257$	− 13.5279i	− 0.843845i	0.906632	−	0.421922i	$-0.138645\pi$
$258$	0	0
$259$	5.90983	0.367219
$260$	0	0
$261$	−2.76393	−0.171083
$262$	0	0
$263$	14.6180i	0.901387i	0.892679	+	0.450693i	$0.148823\pi$
$263$	14.6180i	0.901387i	−0.892679	+	0.450693i	$0.851177\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.0902i	− 1.10710i
$268$	0	0
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	−25.4164	−1.54394	−0.771968	−	0.635661i	$-0.780728\pi$
$271$	−25.4164	−1.54394	−0.771968	+	0.635661i	$0.780728\pi$
$272$	0	0
$273$	− 3.00000i	− 0.181568i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.8541i	0.772328i	0.922430	+	0.386164i	$0.126200\pi$
$277$	12.8541i	0.772328i	−0.922430	+	0.386164i	$0.873800\pi$
$278$	0	0
$279$	−2.47214	−0.148003
$280$	0	0
$281$	−6.09017	−0.363309	−0.181655	−	0.983362i	$-0.558145\pi$
$281$	−6.09017	−0.363309	−0.181655	+	0.983362i	$0.558145\pi$
$282$	0	0
$283$	− 30.1803i	− 1.79403i	−0.441995	−	0.897017i	$-0.645729\pi$
$283$	− 30.1803i	− 1.79403i	0.441995	−	0.897017i	$-0.354271\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.79837i	0.342267i
$288$	0	0
$289$	16.4164	0.965671
$290$	0	0
$291$	−9.70820	−0.569105
$292$	0	0
$293$	22.0902i	1.29052i	0.763962	+	0.645261i	$0.223251\pi$
$293$	22.0902i	1.29052i	−0.763962	+	0.645261i	$0.776749\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.61803i	0.0938879i
$298$	0	0
$299$	23.5623	1.36264
$300$	0	0
$301$	−3.52786	−0.203343
$302$	0	0
$303$	− 12.6525i	− 0.726866i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.7639i	1.18506i	0.805548	+	0.592530i	$0.201871\pi$
$307$	20.7639i	1.18506i	−0.805548	+	0.592530i	$0.798129\pi$
$308$	0	0
$309$	8.56231	0.487093
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	2.29180i	0.129540i	0.997900	+	0.0647700i	$0.0206314\pi$
$313$	2.29180i	0.129540i	−0.997900	+	0.0647700i	$0.979369\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.14590i	0.0643600i	0.999482	+	0.0321800i	$0.0102450\pi$
$317$	1.14590i	0.0643600i	−0.999482	+	0.0321800i	$0.989755\pi$
$318$	0	0
$319$	4.47214	0.250392
$320$	0	0
$321$	6.94427	0.387591
$322$	0	0
$323$	− 4.47214i	− 0.248836i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6.18034i	− 0.341774i
$328$	0	0
$329$	1.00000	0.0551318
$330$	0	0
$331$	2.47214	0.135881	0.0679404	−	0.997689i	$-0.478357\pi$
$331$	2.47214	0.135881	0.0679404	+	0.997689i	$0.478357\pi$
$332$	0	0
$333$	− 9.56231i	− 0.524011i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.8885i	1.08340i	0.840573	+	0.541699i	$0.182219\pi$
$337$	19.8885i	1.08340i	−0.840573	+	0.541699i	$0.817781\pi$
$338$	0	0
$339$	−3.23607	−0.175759
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	− 8.41641i	− 0.454443i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.0689i	− 1.39945i	−0.714412	−	0.699726i	$-0.753306\pi$
$347$	− 26.0689i	− 1.39945i	0.714412	−	0.699726i	$-0.246694\pi$
$348$	0	0
$349$	26.8328	1.43633	0.718164	−	0.695874i	$-0.244983\pi$
$349$	26.8328	1.43633	0.718164	+	0.695874i	$0.244983\pi$
$350$	0	0
$351$	−4.85410	−0.259093
$352$	0	0
$353$	15.7082i	0.836063i	0.908432	+	0.418032i	$0.137280\pi$
$353$	15.7082i	0.836063i	−0.908432	+	0.418032i	$0.862720\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.472136i	0.0249881i
$358$	0	0
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	8.38197i	0.439939i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.58359i	0.239262i	0.992818	+	0.119631i	$0.0381711\pi$
$367$	4.58359i	0.239262i	−0.992818	+	0.119631i	$0.961829\pi$
$368$	0	0
$369$	9.38197	0.488406
$370$	0	0
$371$	3.32624	0.172690
$372$	0	0
$373$	30.5066i	1.57957i	0.613383	+	0.789785i	$0.289808\pi$
$373$	30.5066i	1.57957i	−0.613383	+	0.789785i	$0.710192\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.4164i	0.690980i
$378$	0	0
$379$	−15.9787	−0.820771	−0.410386	−	0.911912i	$-0.634606\pi$
$379$	−15.9787	−0.820771	−0.410386	+	0.911912i	$0.634606\pi$
$380$	0	0
$381$	11.4164	0.584880
$382$	0	0
$383$	− 37.0902i	− 1.89522i	−0.319431	−	0.947610i	$-0.603492\pi$
$383$	− 37.0902i	− 1.89522i	0.319431	−	0.947610i	$-0.396508\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.70820i	0.290164i
$388$	0	0
$389$	−12.7639	−0.647157	−0.323579	−	0.946201i	$-0.604886\pi$
$389$	−12.7639	−0.647157	−0.323579	+	0.946201i	$0.604886\pi$
$390$	0	0
$391$	−3.70820	−0.187532
$392$	0	0
$393$	3.90983i	0.197225i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.2705i	− 0.866782i	−0.901206	−	0.433391i	$-0.857317\pi$
$397$	− 17.2705i	− 0.866782i	0.901206	−	0.433391i	$-0.142683\pi$
$398$	0	0
$399$	−3.61803	−0.181128
$400$	0	0
$401$	2.32624	0.116167	0.0580834	−	0.998312i	$-0.481501\pi$
$401$	2.32624	0.116167	0.0580834	+	0.998312i	$0.481501\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.4721i	0.766925i
$408$	0	0
$409$	14.1459	0.699470	0.349735	−	0.936849i	$-0.386272\pi$
$409$	14.1459	0.699470	0.349735	+	0.936849i	$0.386272\pi$
$410$	0	0
$411$	3.05573	0.150728
$412$	0	0
$413$	8.09017i	0.398091i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.5623i	− 0.615179i
$418$	0	0
$419$	11.0557	0.540108	0.270054	−	0.962845i	$-0.412958\pi$
$419$	11.0557	0.540108	0.270054	+	0.962845i	$0.412958\pi$
$420$	0	0
$421$	18.1803	0.886056	0.443028	−	0.896508i	$-0.353904\pi$
$421$	18.1803	0.886056	0.443028	+	0.896508i	$0.353904\pi$
$422$	0	0
$423$	− 1.61803i	− 0.0786715i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	7.85410	0.379200
$430$	0	0
$431$	−15.8197	−0.762006	−0.381003	−	0.924574i	$-0.624421\pi$
$431$	−15.8197	−0.762006	−0.381003	+	0.924574i	$0.624421\pi$
$432$	0	0
$433$	17.4164i	0.836979i	0.908222	+	0.418490i	$0.137440\pi$
$433$	17.4164i	0.836979i	−0.908222	+	0.418490i	$0.862560\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 28.4164i	− 1.35934i
$438$	0	0
$439$	1.70820	0.0815281	0.0407641	−	0.999169i	$-0.487021\pi$
$439$	1.70820	0.0815281	0.0407641	+	0.999169i	$0.487021\pi$
$440$	0	0
$441$	−6.61803	−0.315144
$442$	0	0
$443$	− 13.5967i	− 0.646001i	−0.946399	−	0.323000i	$-0.895308\pi$
$443$	− 13.5967i	− 0.646001i	0.946399	−	0.323000i	$-0.104692\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 13.4164i	− 0.634574i
$448$	0	0
$449$	−28.2148	−1.33154	−0.665769	−	0.746158i	$-0.731896\pi$
$449$	−28.2148	−1.33154	−0.665769	+	0.746158i	$0.731896\pi$
$450$	0	0
$451$	−15.1803	−0.714814
$452$	0	0
$453$	10.2918i	0.483551i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.8328i	1.81652i	0.418404	+	0.908261i	$0.362590\pi$
$457$	38.8328i	1.81652i	−0.418404	+	0.908261i	$0.637410\pi$
$458$	0	0
$459$	0.763932	0.0356573
$460$	0	0
$461$	38.1803	1.77824	0.889118	−	0.457678i	$-0.151319\pi$
$461$	38.1803	1.77824	0.889118	+	0.457678i	$0.151319\pi$
$462$	0	0
$463$	31.7771i	1.47681i	0.674359	+	0.738403i	$0.264420\pi$
$463$	31.7771i	1.47681i	−0.674359	+	0.738403i	$0.735580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.2361i	0.705041i	0.935804	+	0.352521i	$0.114675\pi$
$467$	15.2361i	0.705041i	−0.935804	+	0.352521i	$0.885325\pi$
$468$	0	0
$469$	−2.29180	−0.105825
$470$	0	0
$471$	10.9443	0.504285
$472$	0	0
$473$	− 9.23607i	− 0.424675i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.38197i	− 0.246423i
$478$	0	0
$479$	8.29180	0.378862	0.189431	−	0.981894i	$-0.439336\pi$
$479$	8.29180	0.378862	0.189431	+	0.981894i	$0.439336\pi$
$480$	0	0
$481$	−46.4164	−2.11641
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17.3262i	− 0.785127i	−0.919725	−	0.392563i	$-0.871588\pi$
$487$	− 17.3262i	− 0.785127i	0.919725	−	0.392563i	$-0.128412\pi$
$488$	0	0
$489$	4.94427	0.223588
$490$	0	0
$491$	18.9787	0.856497	0.428249	−	0.903661i	$-0.359131\pi$
$491$	18.9787	0.856497	0.428249	+	0.903661i	$0.359131\pi$
$492$	0	0
$493$	− 2.11146i	− 0.0950952i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.18034i	− 0.0978016i
$498$	0	0
$499$	−5.72949	−0.256487	−0.128244	−	0.991743i	$-0.540934\pi$
$499$	−5.72949	−0.256487	−0.128244	+	0.991743i	$0.540934\pi$
$500$	0	0
$501$	−7.85410	−0.350895
$502$	0	0
$503$	13.0344i	0.581177i	0.956848	+	0.290589i	$0.0938511\pi$
$503$	13.0344i	0.581177i	−0.956848	+	0.290589i	$0.906149\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.5623i	0.469088i
$508$	0	0
$509$	−30.6525	−1.35865	−0.679324	−	0.733839i	$-0.737727\pi$
$509$	−30.6525	−1.35865	−0.679324	+	0.733839i	$0.737727\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	5.85410i	0.258465i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.61803i	0.115141i
$518$	0	0
$519$	−14.6180	−0.641660
$520$	0	0
$521$	−38.4508	−1.68456	−0.842281	−	0.539038i	$-0.818788\pi$
$521$	−38.4508	−1.68456	−0.842281	+	0.539038i	$0.818788\pi$
$522$	0	0
$523$	− 29.5279i	− 1.29116i	−0.763691	−	0.645582i	$-0.776615\pi$
$523$	− 29.5279i	− 1.29116i	0.763691	−	0.645582i	$-0.223385\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.88854i	− 0.0822663i
$528$	0	0
$529$	−0.562306	−0.0244481
$530$	0	0
$531$	13.0902	0.568065
$532$	0	0
$533$	− 45.5410i	− 1.97260i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 22.0344i	− 0.950856i
$538$	0	0
$539$	10.7082	0.461235
$540$	0	0
$541$	45.4164	1.95260	0.976302	−	0.216413i	$-0.0694357\pi$
$541$	45.4164	1.95260	0.976302	+	0.216413i	$0.0694357\pi$
$542$	0	0
$543$	− 25.4164i	− 1.09072i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23.1246i	0.988737i	0.869252	+	0.494369i	$0.164601\pi$
$547$	23.1246i	0.988737i	−0.869252	+	0.494369i	$0.835399\pi$
$548$	0	0
$549$	9.70820	0.414336
$550$	0	0
$551$	16.1803	0.689306
$552$	0	0
$553$	− 8.29180i	− 0.352603i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11.6180i	− 0.492272i	−0.969235	−	0.246136i	$-0.920839\pi$
$557$	− 11.6180i	− 0.492272i	0.969235	−	0.246136i	$-0.0791610\pi$
$558$	0	0
$559$	27.7082	1.17193
$560$	0	0
$561$	−1.23607	−0.0521868
$562$	0	0
$563$	− 2.94427i	− 0.124086i	−0.998073	−	0.0620431i	$-0.980238\pi$
$563$	− 2.94427i	− 0.124086i	0.998073	−	0.0620431i	$-0.0197616\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.618034i	− 0.0259550i
$568$	0	0
$569$	14.6738	0.615156	0.307578	−	0.951523i	$-0.400481\pi$
$569$	14.6738	0.615156	0.307578	+	0.951523i	$0.400481\pi$
$570$	0	0
$571$	26.2148	1.09705	0.548527	−	0.836133i	$-0.315189\pi$
$571$	26.2148	1.09705	0.548527	+	0.836133i	$0.315189\pi$
$572$	0	0
$573$	− 16.9443i	− 0.707857i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.7082i	1.40329i	0.712526	+	0.701645i	$0.247551\pi$
$577$	33.7082i	1.40329i	−0.712526	+	0.701645i	$0.752449\pi$
$578$	0	0
$579$	−4.29180	−0.178361
$580$	0	0
$581$	9.23607	0.383177
$582$	0	0
$583$	8.70820i	0.360657i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.8197i	− 0.652947i	−0.945206	−	0.326474i	$-0.894140\pi$
$587$	− 15.8197i	− 0.652947i	0.945206	−	0.326474i	$-0.105860\pi$
$588$	0	0
$589$	14.4721	0.596314
$590$	0	0
$591$	−6.94427	−0.285649
$592$	0	0
$593$	1.63932i	0.0673188i	0.999433	+	0.0336594i	$0.0107161\pi$
$593$	1.63932i	0.0673188i	−0.999433	+	0.0336594i	$0.989284\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.18034i	0.252944i
$598$	0	0
$599$	48.5410	1.98333	0.991666	−	0.128834i	$-0.0411235\pi$
$599$	48.5410	1.98333	0.991666	+	0.128834i	$0.0411235\pi$
$600$	0	0
$601$	−32.2705	−1.31634	−0.658171	−	0.752869i	$-0.728670\pi$
$601$	−32.2705	−1.31634	−0.658171	+	0.752869i	$0.728670\pi$
$602$	0	0
$603$	3.70820i	0.151010i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.6869i	0.839656i	0.907604	+	0.419828i	$0.137910\pi$
$607$	20.6869i	0.839656i	−0.907604	+	0.419828i	$0.862090\pi$
$608$	0	0
$609$	−1.70820	−0.0692199
$610$	0	0
$611$	−7.85410	−0.317743
$612$	0	0
$613$	− 22.3820i	− 0.903999i	−0.892018	−	0.452000i	$-0.850711\pi$
$613$	− 22.3820i	− 0.903999i	0.892018	−	0.452000i	$-0.149289\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 29.3050i	− 1.17977i	−0.807486	−	0.589886i	$-0.799172\pi$
$617$	− 29.3050i	− 1.17977i	0.807486	−	0.589886i	$-0.200828\pi$
$618$	0	0
$619$	−18.0902	−0.727105	−0.363553	−	0.931574i	$-0.618436\pi$
$619$	−18.0902	−0.727105	−0.363553	+	0.931574i	$0.618436\pi$
$620$	0	0
$621$	4.85410	0.194788
$622$	0	0
$623$	− 11.1803i	− 0.447931i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 9.47214i	− 0.378281i
$628$	0	0
$629$	7.30495	0.291267
$630$	0	0
$631$	1.81966	0.0724395	0.0362198	−	0.999344i	$-0.488468\pi$
$631$	1.81966	0.0724395	0.0362198	+	0.999344i	$0.488468\pi$
$632$	0	0
$633$	− 17.2705i	− 0.686441i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	32.1246i	1.27282i
$638$	0	0
$639$	−3.52786	−0.139560
$640$	0	0
$641$	−1.74265	−0.0688304	−0.0344152	−	0.999408i	$-0.510957\pi$
$641$	−1.74265	−0.0688304	−0.0344152	+	0.999408i	$0.510957\pi$
$642$	0	0
$643$	7.05573i	0.278251i	0.990275	+	0.139125i	$0.0444291\pi$
$643$	7.05573i	0.278251i	−0.990275	+	0.139125i	$0.955571\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.09017i	− 0.200115i	−0.994982	−	0.100058i	$-0.968097\pi$
$647$	− 5.09017i	− 0.200115i	0.994982	−	0.100058i	$-0.0319027\pi$
$648$	0	0
$649$	−21.1803	−0.831401
$650$	0	0
$651$	−1.52786	−0.0598817
$652$	0	0
$653$	13.1459i	0.514439i	0.966353	+	0.257219i	$0.0828063\pi$
$653$	13.1459i	0.514439i	−0.966353	+	0.257219i	$0.917194\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 12.9443i	− 0.505004i
$658$	0	0
$659$	−1.25735	−0.0489796	−0.0244898	−	0.999700i	$-0.507796\pi$
$659$	−1.25735	−0.0489796	−0.0244898	+	0.999700i	$0.507796\pi$
$660$	0	0
$661$	2.65248	0.103169	0.0515847	−	0.998669i	$-0.483573\pi$
$661$	2.65248	0.103169	0.0515847	+	0.998669i	$0.483573\pi$
$662$	0	0
$663$	− 3.70820i	− 0.144015i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 13.4164i	− 0.519485i
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−15.7082	−0.606408
$672$	0	0
$673$	− 21.5279i	− 0.829838i	−0.909858	−	0.414919i	$-0.863810\pi$
$673$	− 21.5279i	− 0.829838i	0.909858	−	0.414919i	$-0.136190\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.85410i	0.301858i	0.988545	+	0.150929i	$0.0482264\pi$
$677$	7.85410i	0.301858i	−0.988545	+	0.150929i	$0.951774\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−28.1803	−1.07987
$682$	0	0
$683$	24.9443i	0.954466i	0.878777	+	0.477233i	$0.158360\pi$
$683$	24.9443i	0.954466i	−0.878777	+	0.477233i	$0.841640\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.70820i	0.0651720i
$688$	0	0
$689$	−26.1246	−0.995268
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	1.00000i	0.0379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.16718i	0.271476i
$698$	0	0
$699$	−19.4164	−0.734396
$700$	0	0
$701$	−25.2361	−0.953153	−0.476577	−	0.879133i	$-0.658122\pi$
$701$	−25.2361	−0.953153	−0.476577	+	0.879133i	$0.658122\pi$
$702$	0	0
$703$	55.9787i	2.11128i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.81966i	− 0.294089i
$708$	0	0
$709$	2.76393	0.103802	0.0519008	−	0.998652i	$-0.483472\pi$
$709$	2.76393	0.103802	0.0519008	+	0.998652i	$0.483472\pi$
$710$	0	0
$711$	−13.4164	−0.503155
$712$	0	0
$713$	− 12.0000i	− 0.449404i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8.94427i	− 0.334030i
$718$	0	0
$719$	−27.8885	−1.04007	−0.520034	−	0.854146i	$-0.674081\pi$
$719$	−27.8885	−1.04007	−0.520034	+	0.854146i	$0.674081\pi$
$720$	0	0
$721$	5.29180	0.197077
$722$	0	0
$723$	7.14590i	0.265759i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.1033i	1.63570i	0.575430	+	0.817851i	$0.304835\pi$
$727$	44.1033i	1.63570i	−0.575430	+	0.817851i	$0.695165\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−4.36068	−0.161286
$732$	0	0
$733$	− 31.3262i	− 1.15706i	−0.815661	−	0.578530i	$-0.803626\pi$
$733$	− 31.3262i	− 1.15706i	0.815661	−	0.578530i	$-0.196374\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.00000i	− 0.221013i
$738$	0	0
$739$	10.7295	0.394691	0.197345	−	0.980334i	$-0.436768\pi$
$739$	10.7295	0.394691	0.197345	+	0.980334i	$0.436768\pi$
$740$	0	0
$741$	28.4164	1.04390
$742$	0	0
$743$	− 4.85410i	− 0.178080i	−0.996028	−	0.0890399i	$-0.971620\pi$
$743$	− 4.85410i	− 0.178080i	0.996028	−	0.0890399i	$-0.0283799\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 14.9443i	− 0.546782i
$748$	0	0
$749$	4.29180	0.156819
$750$	0	0
$751$	47.5967	1.73683	0.868415	−	0.495838i	$-0.165139\pi$
$751$	47.5967	1.73683	0.868415	+	0.495838i	$0.165139\pi$
$752$	0	0
$753$	20.9443i	0.763252i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.7984i	0.792275i	0.918191	+	0.396138i	$0.129650\pi$
$757$	21.7984i	0.792275i	−0.918191	+	0.396138i	$0.870350\pi$
$758$	0	0
$759$	−7.85410	−0.285086
$760$	0	0
$761$	−10.0344	−0.363748	−0.181874	−	0.983322i	$-0.558216\pi$
$761$	−10.0344	−0.363748	−0.181874	+	0.983322i	$0.558216\pi$
$762$	0	0
$763$	− 3.81966i	− 0.138281i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 63.5410i	− 2.29433i
$768$	0	0
$769$	15.7295	0.567220	0.283610	−	0.958940i	$-0.408468\pi$
$769$	15.7295	0.567220	0.283610	+	0.958940i	$0.408468\pi$
$770$	0	0
$771$	−13.5279	−0.487194
$772$	0	0
$773$	2.94427i	0.105898i	0.998597	+	0.0529491i	$0.0168621\pi$
$773$	2.94427i	0.105898i	−0.998597	+	0.0529491i	$0.983138\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 5.90983i	− 0.212014i
$778$	0	0
$779$	−54.9230	−1.96782
$780$	0	0
$781$	5.70820	0.204256
$782$	0	0
$783$	2.76393i	0.0987749i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 25.4164i	− 0.905997i	−0.891511	−	0.452999i	$-0.850354\pi$
$787$	− 25.4164i	− 0.905997i	0.891511	−	0.452999i	$-0.149646\pi$
$788$	0	0
$789$	14.6180	0.520416
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	− 47.1246i	− 1.67344i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.09017i	0.180303i	0.995928	+	0.0901515i	$0.0287351\pi$
$797$	5.09017i	0.180303i	−0.995928	+	0.0901515i	$0.971265\pi$
$798$	0	0
$799$	1.23607	0.0437289
$800$	0	0
$801$	−18.0902	−0.639185
$802$	0	0
$803$	20.9443i	0.739107i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.4164i	− 0.472280i
$808$	0	0
$809$	−15.9787	−0.561782	−0.280891	−	0.959740i	$-0.590630\pi$
$809$	−15.9787	−0.561782	−0.280891	+	0.959740i	$0.590630\pi$
$810$	0	0
$811$	−53.1033	−1.86471	−0.932355	−	0.361544i	$-0.882250\pi$
$811$	−53.1033	−1.86471	−0.932355	+	0.361544i	$0.882250\pi$
$812$	0	0
$813$	25.4164i	0.891392i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 33.4164i	− 1.16909i
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−16.9443	−0.591359	−0.295680	−	0.955287i	$-0.595546\pi$
$821$	−16.9443	−0.591359	−0.295680	+	0.955287i	$0.595546\pi$
$822$	0	0
$823$	8.03444i	0.280063i	0.990147	+	0.140032i	$0.0447204\pi$
$823$	8.03444i	0.280063i	−0.990147	+	0.140032i	$0.955280\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.5410i	1.27066i	0.772243	+	0.635328i	$0.219135\pi$
$827$	36.5410i	1.27066i	−0.772243	+	0.635328i	$0.780865\pi$
$828$	0	0
$829$	40.2492	1.39791	0.698957	−	0.715164i	$-0.253648\pi$
$829$	40.2492	1.39791	0.698957	+	0.715164i	$0.253648\pi$
$830$	0	0
$831$	12.8541	0.445904
$832$	0	0
$833$	− 5.05573i	− 0.175171i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.47214i	0.0854495i
$838$	0	0
$839$	−16.1803	−0.558607	−0.279304	−	0.960203i	$-0.590104\pi$
$839$	−16.1803	−0.558607	−0.279304	+	0.960203i	$0.590104\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	6.09017i	0.209757i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.18034i	0.177999i
$848$	0	0
$849$	−30.1803	−1.03579
$850$	0	0
$851$	46.4164	1.59113
$852$	0	0
$853$	6.43769i	0.220422i	0.993908	+	0.110211i	$0.0351527\pi$
$853$	6.43769i	0.220422i	−0.993908	+	0.110211i	$0.964847\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.18034i	0.279435i	0.990191	+	0.139718i	$0.0446195\pi$
$857$	8.18034i	0.279435i	−0.990191	+	0.139718i	$0.955381\pi$
$858$	0	0
$859$	0.978714	0.0333933	0.0166966	−	0.999861i	$-0.494685\pi$
$859$	0.978714	0.0333933	0.0166966	+	0.999861i	$0.494685\pi$
$860$	0	0
$861$	5.79837	0.197608
$862$	0	0
$863$	2.90983i	0.0990518i	0.998773	+	0.0495259i	$0.0157710\pi$
$863$	2.90983i	0.0990518i	−0.998773	+	0.0495259i	$0.984229\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.4164i	− 0.557530i
$868$	0	0
$869$	21.7082	0.736400
$870$	0	0
$871$	18.0000	0.609907
$872$	0	0
$873$	9.70820i	0.328573i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.72949i	0.261006i	0.991448	+	0.130503i	$0.0416593\pi$
$877$	7.72949i	0.261006i	−0.991448	+	0.130503i	$0.958341\pi$
$878$	0	0
$879$	22.0902	0.745083
$880$	0	0
$881$	−8.72949	−0.294104	−0.147052	−	0.989129i	$-0.546978\pi$
$881$	−8.72949	−0.294104	−0.147052	+	0.989129i	$0.546978\pi$
$882$	0	0
$883$	− 38.4721i	− 1.29469i	−0.762197	−	0.647345i	$-0.775879\pi$
$883$	− 38.4721i	− 1.29469i	0.762197	−	0.647345i	$-0.224121\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.9098i	1.17216i	0.810254	+	0.586079i	$0.199329\pi$
$887$	34.9098i	1.17216i	−0.810254	+	0.586079i	$0.800671\pi$
$888$	0	0
$889$	7.05573	0.236642
$890$	0	0
$891$	1.61803	0.0542062
$892$	0	0
$893$	9.47214i	0.316973i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 23.5623i	− 0.786722i
$898$	0	0
$899$	6.83282	0.227887
$900$	0	0
$901$	4.11146	0.136972
$902$	0	0
$903$	3.52786i	0.117400i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.1246i	0.435796i	0.975972	+	0.217898i	$0.0699199\pi$
$907$	13.1246i	0.435796i	−0.975972	+	0.217898i	$0.930080\pi$
$908$	0	0
$909$	−12.6525	−0.419656
$910$	0	0
$911$	35.8885	1.18904	0.594520	−	0.804081i	$-0.297342\pi$
$911$	35.8885	1.18904	0.594520	+	0.804081i	$0.297342\pi$
$912$	0	0
$913$	24.1803i	0.800252i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.41641i	0.0797968i
$918$	0	0
$919$	−29.5967	−0.976307	−0.488153	−	0.872758i	$-0.662329\pi$
$919$	−29.5967	−0.976307	−0.488153	+	0.872758i	$0.662329\pi$
$920$	0	0
$921$	20.7639	0.684195
$922$	0	0
$923$	17.1246i	0.563663i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.56231i	− 0.281223i
$928$	0	0
$929$	30.4508	0.999060	0.499530	−	0.866297i	$-0.333506\pi$
$929$	30.4508	0.999060	0.499530	+	0.866297i	$0.333506\pi$
$930$	0	0
$931$	38.7426	1.26974
$932$	0	0
$933$	12.0000i	0.392862i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.1246i	0.886122i	0.896491	+	0.443061i	$0.146108\pi$
$937$	27.1246i	0.886122i	−0.896491	+	0.443061i	$0.853892\pi$
$938$	0	0
$939$	2.29180	0.0747899
$940$	0	0
$941$	17.1246	0.558246	0.279123	−	0.960255i	$-0.409956\pi$
$941$	17.1246	0.558246	0.279123	+	0.960255i	$0.409956\pi$
$942$	0	0
$943$	45.5410i	1.48302i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 29.8885i	− 0.971247i	−0.874168	−	0.485624i	$-0.838593\pi$
$947$	− 29.8885i	− 0.971247i	0.874168	−	0.485624i	$-0.161407\pi$
$948$	0	0
$949$	−62.8328	−2.03964
$950$	0	0
$951$	1.14590	0.0371583
$952$	0	0
$953$	60.1803i	1.94943i	0.223446	+	0.974716i	$0.428269\pi$
$953$	60.1803i	1.94943i	−0.223446	+	0.974716i	$0.571731\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.47214i	− 0.144564i
$958$	0	0
$959$	1.88854	0.0609843
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	− 6.94427i	− 0.223776i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.3262i	0.428543i	0.976774	+	0.214271i	$0.0687377\pi$
$967$	13.3262i	0.428543i	−0.976774	+	0.214271i	$0.931262\pi$
$968$	0	0
$969$	−4.47214	−0.143666
$970$	0	0
$971$	2.02129	0.0648662	0.0324331	−	0.999474i	$-0.489674\pi$
$971$	2.02129	0.0648662	0.0324331	+	0.999474i	$0.489674\pi$
$972$	0	0
$973$	− 7.76393i	− 0.248900i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.58359i	− 0.146642i	−0.997308	−	0.0733211i	$-0.976640\pi$
$977$	− 4.58359i	− 0.146642i	0.997308	−	0.0733211i	$-0.0233598\pi$
$978$	0	0
$979$	29.2705	0.935490
$980$	0	0
$981$	−6.18034	−0.197323
$982$	0	0
$983$	− 24.4508i	− 0.779861i	−0.920844	−	0.389930i	$-0.872499\pi$
$983$	− 24.4508i	− 0.779861i	0.920844	−	0.389930i	$-0.127501\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.00000i	− 0.0318304i
$988$	0	0
$989$	−27.7082	−0.881070
$990$	0	0
$991$	−42.0000	−1.33417	−0.667087	−	0.744980i	$-0.732459\pi$
$991$	−42.0000	−1.33417	−0.667087	+	0.744980i	$0.732459\pi$
$992$	0	0
$993$	− 2.47214i	− 0.0784509i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22.1459i	− 0.701368i	−0.936494	−	0.350684i	$-0.885949\pi$
$997$	− 22.1459i	− 0.701368i	0.936494	−	0.350684i	$-0.114051\pi$
$998$	0	0
$999$	−9.56231	−0.302538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.h.1249.1		4
4.3	odd	2		750.2.c.d.499.2		4
5.2	odd	4		6000.2.a.f.1.2		2
5.3	odd	4		6000.2.a.w.1.1		2
5.4	even	2	inner	6000.2.f.h.1249.4		4
12.11	even	2		2250.2.c.d.1999.4		4
20.3	even	4		750.2.a.b.1.2	✓	2
20.7	even	4		750.2.a.g.1.1	yes	2
20.19	odd	2		750.2.c.d.499.3		4
60.23	odd	4		2250.2.a.l.1.2		2
60.47	odd	4		2250.2.a.e.1.1		2
60.59	even	2		2250.2.c.d.1999.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.b.1.2	✓	2	20.3	even	4
750.2.a.g.1.1	yes	2	20.7	even	4
750.2.c.d.499.2		4	4.3	odd	2
750.2.c.d.499.3		4	20.19	odd	2
2250.2.a.e.1.1		2	60.47	odd	4
2250.2.a.l.1.2		2	60.23	odd	4
2250.2.c.d.1999.1		4	60.59	even	2
2250.2.c.d.1999.4		4	12.11	even	2
6000.2.a.f.1.2		2	5.2	odd	4
6000.2.a.w.1.1		2	5.3	odd	4
6000.2.f.h.1249.1		4	1.1	even	1	trivial
6000.2.f.h.1249.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.618034i q^{7} -1.00000 q^{9} +1.61803 q^{11} +4.85410i q^{13} -0.763932i q^{17} +5.85410 q^{19} -0.618034 q^{21} -4.85410i q^{23} +1.00000i q^{27} +2.76393 q^{29} +2.47214 q^{31} -1.61803i q^{33} +9.56231i q^{37} +4.85410 q^{39} -9.38197 q^{41} -5.70820i q^{43} +1.61803i q^{47} +6.61803 q^{49} -0.763932 q^{51} +5.38197i q^{53} -5.85410i q^{57} -13.0902 q^{59} -9.70820 q^{61} +0.618034i q^{63} -3.70820i q^{67} -4.85410 q^{69} +3.52786 q^{71} +12.9443i q^{73} -1.00000i q^{77} +13.4164 q^{79} +1.00000 q^{81} +14.9443i q^{83} -2.76393i q^{87} +18.0902 q^{89} +3.00000 q^{91} -2.47214i q^{93} -9.70820i q^{97} -1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 2 q^{11} + 10 q^{19} + 2 q^{21} + 20 q^{29} - 8 q^{31} + 6 q^{39} - 42 q^{41} + 22 q^{49} - 12 q^{51} - 30 q^{59} - 12 q^{61} - 6 q^{69} + 32 q^{71} + 4 q^{81} + 50 q^{89} + 12 q^{91} - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 2 * q^11 + 10 * q^19 + 2 * q^21 + 20 * q^29 - 8 * q^31 + 6 * q^39 - 42 * q^41 + 22 * q^49 - 12 * q^51 - 30 * q^59 - 12 * q^61 - 6 * q^69 + 32 * q^71 + 4 * q^81 + 50 * q^89 + 12 * q^91 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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