LMFDB - Embedded newform 6000.2.f.m.1249.3

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,-4,0,16,0,0,0,0,0,0,0,-30,0,-4] 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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1500)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.m.1249.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.23607i q^{7} -1.00000 q^{9} +4.00000 q^{11} +4.47214i q^{13} -6.85410i q^{17} -8.61803 q^{19} +1.23607 q^{21} -2.38197i q^{23} -1.00000i q^{27} +7.70820 q^{29} -2.09017 q^{31} +4.00000i q^{33} +9.23607i q^{37} -4.47214 q^{39} +8.94427 q^{41} +4.00000i q^{43} -6.32624i q^{47} +5.47214 q^{49} +6.85410 q^{51} -1.09017i q^{53} -8.61803i q^{57} -6.00000 q^{59} -2.14590 q^{61} +1.23607i q^{63} -3.23607i q^{67} +2.38197 q^{69} +4.47214 q^{71} -13.7082i q^{73} -4.94427i q^{77} +8.32624 q^{79} +1.00000 q^{81} -3.56231i q^{83} +7.70820i q^{87} +11.7082 q^{89} +5.52786 q^{91} -2.09017i q^{93} +5.23607i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 16 q^{11} - 30 q^{19} - 4 q^{21} + 4 q^{29} + 14 q^{31} + 4 q^{49} + 14 q^{51} - 24 q^{59} - 22 q^{61} + 14 q^{69} + 2 q^{79} + 4 q^{81} + 20 q^{89} + 40 q^{91} - 16 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 16 * q^11 - 30 * q^19 - 4 * q^21 + 4 * q^29 + 14 * q^31 + 4 * q^49 + 14 * q^51 - 24 * q^59 - 22 * q^61 + 14 * q^69 + 2 * q^79 + 4 * q^81 + 20 * q^89 + 40 * q^91 - 16 * 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
$7$	− 1.23607i	− 0.467190i	−0.972334	−	0.233595i	$-0.924951\pi$
$7$	− 1.23607i	− 0.467190i	0.972334	−	0.233595i	$-0.0750489\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.85410i	− 1.66236i	−0.556001	−	0.831182i	$-0.687665\pi$
$17$	− 6.85410i	− 1.66236i	0.556001	−	0.831182i	$-0.312335\pi$
$18$	0	0
$19$	−8.61803	−1.97711	−0.988556	−	0.150852i	$-0.951798\pi$
$19$	−8.61803	−1.97711	−0.988556	+	0.150852i	$0.951798\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	− 2.38197i	− 0.496674i	−0.968674	−	0.248337i	$-0.920116\pi$
$23$	− 2.38197i	− 0.496674i	0.968674	−	0.248337i	$-0.0798841\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	7.70820	1.43138	0.715689	−	0.698419i	$-0.246113\pi$
$29$	7.70820	1.43138	0.715689	+	0.698419i	$0.246113\pi$
$30$	0	0
$31$	−2.09017	−0.375406	−0.187703	−	0.982226i	$-0.560104\pi$
$31$	−2.09017	−0.375406	−0.187703	+	0.982226i	$0.560104\pi$
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.23607i	1.51840i	0.650857	+	0.759200i	$0.274410\pi$
$37$	9.23607i	1.51840i	−0.650857	+	0.759200i	$0.725590\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	8.94427	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.32624i	− 0.922777i	−0.887198	−	0.461388i	$-0.847352\pi$
$47$	− 6.32624i	− 0.922777i	0.887198	−	0.461388i	$-0.152648\pi$
$48$	0	0
$49$	5.47214	0.781734
$50$	0	0
$51$	6.85410	0.959766
$52$	0	0
$53$	− 1.09017i	− 0.149746i	−0.997193	−	0.0748732i	$-0.976145\pi$
$53$	− 1.09017i	− 0.149746i	0.997193	−	0.0748732i	$-0.0238552\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.61803i	− 1.14149i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−2.14590	−0.274754	−0.137377	−	0.990519i	$-0.543867\pi$
$61$	−2.14590	−0.274754	−0.137377	+	0.990519i	$0.543867\pi$
$62$	0	0
$63$	1.23607i	0.155730i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.23607i	− 0.395349i	−0.980268	−	0.197674i	$-0.936661\pi$
$67$	− 3.23607i	− 0.395349i	0.980268	−	0.197674i	$-0.0633388\pi$
$68$	0	0
$69$	2.38197	0.286755
$70$	0	0
$71$	4.47214	0.530745	0.265372	−	0.964146i	$-0.414505\pi$
$71$	4.47214	0.530745	0.265372	+	0.964146i	$0.414505\pi$
$72$	0	0
$73$	− 13.7082i	− 1.60442i	−0.597039	−	0.802212i	$-0.703656\pi$
$73$	− 13.7082i	− 1.60442i	0.597039	−	0.802212i	$-0.296344\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.94427i	− 0.563452i
$78$	0	0
$79$	8.32624	0.936775	0.468387	−	0.883523i	$-0.344835\pi$
$79$	8.32624	0.936775	0.468387	+	0.883523i	$0.344835\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 3.56231i	− 0.391014i	−0.980702	−	0.195507i	$-0.937365\pi$
$83$	− 3.56231i	− 0.391014i	0.980702	−	0.195507i	$-0.0626352\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.70820i	0.826406i
$88$	0	0
$89$	11.7082	1.24107	0.620534	−	0.784180i	$-0.286916\pi$
$89$	11.7082	1.24107	0.620534	+	0.784180i	$0.286916\pi$
$90$	0	0
$91$	5.52786	0.579478
$92$	0	0
$93$	− 2.09017i	− 0.216741i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.23607i	0.531642i	0.964022	+	0.265821i	$0.0856430\pi$
$97$	5.23607i	0.531642i	−0.964022	+	0.265821i	$0.914357\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−0.291796	−0.0290348	−0.0145174	−	0.999895i	$-0.504621\pi$
$101$	−0.291796	−0.0290348	−0.0145174	+	0.999895i	$0.504621\pi$
$102$	0	0
$103$	− 14.6525i	− 1.44375i	−0.692023	−	0.721876i	$-0.743280\pi$
$103$	− 14.6525i	− 1.44375i	0.692023	−	0.721876i	$-0.256720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.5623i	1.02110i	0.859849	+	0.510548i	$0.170558\pi$
$107$	10.5623i	1.02110i	−0.859849	+	0.510548i	$0.829442\pi$
$108$	0	0
$109$	16.2705	1.55843	0.779216	−	0.626755i	$-0.215617\pi$
$109$	16.2705	1.55843	0.779216	+	0.626755i	$0.215617\pi$
$110$	0	0
$111$	−9.23607	−0.876649
$112$	0	0
$113$	− 7.85410i	− 0.738852i	−0.929260	−	0.369426i	$-0.879554\pi$
$113$	− 7.85410i	− 0.738852i	0.929260	−	0.369426i	$-0.120446\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.47214i	− 0.413449i
$118$	0	0
$119$	−8.47214	−0.776639
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	8.94427i	0.806478i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.4721i	− 1.63914i	−0.572981	−	0.819569i	$-0.694213\pi$
$127$	− 18.4721i	− 1.63914i	0.572981	−	0.819569i	$-0.305787\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−0.944272	−0.0825014	−0.0412507	−	0.999149i	$-0.513134\pi$
$131$	−0.944272	−0.0825014	−0.0412507	+	0.999149i	$0.513134\pi$
$132$	0	0
$133$	10.6525i	0.923687i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9443i	0.935032i	0.883985	+	0.467516i	$0.154851\pi$
$137$	10.9443i	0.935032i	−0.883985	+	0.467516i	$0.845149\pi$
$138$	0	0
$139$	−9.32624	−0.791041	−0.395521	−	0.918457i	$-0.629436\pi$
$139$	−9.32624	−0.791041	−0.395521	+	0.918457i	$0.629436\pi$
$140$	0	0
$141$	6.32624	0.532765
$142$	0	0
$143$	17.8885i	1.49592i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.47214i	0.451334i
$148$	0	0
$149$	−11.4164	−0.935269	−0.467634	−	0.883922i	$-0.654894\pi$
$149$	−11.4164	−0.935269	−0.467634	+	0.883922i	$0.654894\pi$
$150$	0	0
$151$	5.85410	0.476400	0.238200	−	0.971216i	$-0.423443\pi$
$151$	5.85410	0.476400	0.238200	+	0.971216i	$0.423443\pi$
$152$	0	0
$153$	6.85410i	0.554121i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.18034i	0.333627i	0.985988	+	0.166814i	$0.0533479\pi$
$157$	4.18034i	0.333627i	−0.985988	+	0.166814i	$0.946652\pi$
$158$	0	0
$159$	1.09017	0.0864561
$160$	0	0
$161$	−2.94427	−0.232041
$162$	0	0
$163$	8.29180i	0.649464i	0.945806	+	0.324732i	$0.105274\pi$
$163$	8.29180i	0.649464i	−0.945806	+	0.324732i	$0.894726\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.3262i	1.41813i	0.705145	+	0.709063i	$0.250882\pi$
$167$	18.3262i	1.41813i	−0.705145	+	0.709063i	$0.749118\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	8.61803	0.659038
$172$	0	0
$173$	5.05573i	0.384380i	0.981358	+	0.192190i	$0.0615590\pi$
$173$	5.05573i	0.384380i	−0.981358	+	0.192190i	$0.938441\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	1.23607	0.0923881	0.0461940	−	0.998932i	$-0.485291\pi$
$179$	1.23607	0.0923881	0.0461940	+	0.998932i	$0.485291\pi$
$180$	0	0
$181$	11.3820	0.846015	0.423007	−	0.906126i	$-0.360974\pi$
$181$	11.3820	0.846015	0.423007	+	0.906126i	$0.360974\pi$
$182$	0	0
$183$	− 2.14590i	− 0.158629i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 27.4164i	− 2.00489i
$188$	0	0
$189$	−1.23607	−0.0899107
$190$	0	0
$191$	21.1246	1.52852	0.764262	−	0.644906i	$-0.223104\pi$
$191$	21.1246	1.52852	0.764262	+	0.644906i	$0.223104\pi$
$192$	0	0
$193$	− 19.4164i	− 1.39762i	−0.715306	−	0.698812i	$-0.753712\pi$
$193$	− 19.4164i	− 1.39762i	0.715306	−	0.698812i	$-0.246288\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.0902i	− 1.57386i	−0.617043	−	0.786930i	$-0.711669\pi$
$197$	− 22.0902i	− 1.57386i	0.617043	−	0.786930i	$-0.288331\pi$
$198$	0	0
$199$	6.09017	0.431721	0.215860	−	0.976424i	$-0.430744\pi$
$199$	6.09017	0.431721	0.215860	+	0.976424i	$0.430744\pi$
$200$	0	0
$201$	3.23607	0.228255
$202$	0	0
$203$	− 9.52786i	− 0.668725i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.38197i	0.165558i
$208$	0	0
$209$	−34.4721	−2.38449
$210$	0	0
$211$	14.2705	0.982422	0.491211	−	0.871040i	$-0.336554\pi$
$211$	14.2705	0.982422	0.491211	+	0.871040i	$0.336554\pi$
$212$	0	0
$213$	4.47214i	0.306426i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.58359i	0.175386i
$218$	0	0
$219$	13.7082	0.926315
$220$	0	0
$221$	30.6525	2.06191
$222$	0	0
$223$	− 7.23607i	− 0.484563i	−0.970206	−	0.242281i	$-0.922104\pi$
$223$	− 7.23607i	− 0.484563i	0.970206	−	0.242281i	$-0.0778957\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.79837i	0.650341i	0.945655	+	0.325170i	$0.105422\pi$
$227$	9.79837i	0.650341i	−0.945655	+	0.325170i	$0.894578\pi$
$228$	0	0
$229$	−2.38197	−0.157405	−0.0787024	−	0.996898i	$-0.525078\pi$
$229$	−2.38197	−0.157405	−0.0787024	+	0.996898i	$0.525078\pi$
$230$	0	0
$231$	4.94427	0.325309
$232$	0	0
$233$	27.8885i	1.82704i	0.406795	+	0.913520i	$0.366646\pi$
$233$	27.8885i	1.82704i	−0.406795	+	0.913520i	$0.633354\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.32624i	0.540847i
$238$	0	0
$239$	8.65248	0.559682	0.279841	−	0.960046i	$-0.409718\pi$
$239$	8.65248	0.559682	0.279841	+	0.960046i	$0.409718\pi$
$240$	0	0
$241$	14.3820	0.926424	0.463212	−	0.886248i	$-0.346697\pi$
$241$	14.3820	0.926424	0.463212	+	0.886248i	$0.346697\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 38.5410i	− 2.45231i
$248$	0	0
$249$	3.56231	0.225752
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	− 9.52786i	− 0.599012i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.20163i	− 0.199712i	−0.995002	−	0.0998560i	$-0.968162\pi$
$257$	− 3.20163i	− 0.199712i	0.995002	−	0.0998560i	$-0.0318382\pi$
$258$	0	0
$259$	11.4164	0.709381
$260$	0	0
$261$	−7.70820	−0.477126
$262$	0	0
$263$	− 24.0902i	− 1.48546i	−0.669589	−	0.742732i	$-0.733530\pi$
$263$	− 24.0902i	− 1.48546i	0.669589	−	0.742732i	$-0.266470\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.7082i	0.716530i
$268$	0	0
$269$	19.1246	1.16605	0.583024	−	0.812455i	$-0.301869\pi$
$269$	19.1246	1.16605	0.583024	+	0.812455i	$0.301869\pi$
$270$	0	0
$271$	28.5623	1.73504	0.867518	−	0.497405i	$-0.165714\pi$
$271$	28.5623	1.73504	0.867518	+	0.497405i	$0.165714\pi$
$272$	0	0
$273$	5.52786i	0.334562i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.7639i	0.646742i	0.946272	+	0.323371i	$0.104816\pi$
$277$	10.7639i	0.646742i	−0.946272	+	0.323371i	$0.895184\pi$
$278$	0	0
$279$	2.09017	0.125135
$280$	0	0
$281$	3.23607	0.193048	0.0965238	−	0.995331i	$-0.469228\pi$
$281$	3.23607	0.193048	0.0965238	+	0.995331i	$0.469228\pi$
$282$	0	0
$283$	− 32.0689i	− 1.90630i	−0.302502	−	0.953149i	$-0.597822\pi$
$283$	− 32.0689i	− 1.90630i	0.302502	−	0.953149i	$-0.402178\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11.0557i	− 0.652599i
$288$	0	0
$289$	−29.9787	−1.76345
$290$	0	0
$291$	−5.23607	−0.306944
$292$	0	0
$293$	− 17.3262i	− 1.01221i	−0.862472	−	0.506105i	$-0.831085\pi$
$293$	− 17.3262i	− 1.01221i	0.862472	−	0.506105i	$-0.168915\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	10.6525	0.616049
$300$	0	0
$301$	4.94427	0.284983
$302$	0	0
$303$	− 0.291796i	− 0.0167632i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.4721i	1.85328i	0.375947	+	0.926641i	$0.377318\pi$
$307$	32.4721i	1.85328i	−0.375947	+	0.926641i	$0.622682\pi$
$308$	0	0
$309$	14.6525	0.833550
$310$	0	0
$311$	32.1803	1.82478	0.912390	−	0.409322i	$-0.134235\pi$
$311$	32.1803	1.82478	0.912390	+	0.409322i	$0.134235\pi$
$312$	0	0
$313$	− 2.58359i	− 0.146033i	−0.997331	−	0.0730166i	$-0.976737\pi$
$313$	− 2.58359i	− 0.146033i	0.997331	−	0.0730166i	$-0.0232626\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	0	0
$319$	30.8328	1.72631
$320$	0	0
$321$	−10.5623	−0.589530
$322$	0	0
$323$	59.0689i	3.28668i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.2705i	0.899761i
$328$	0	0
$329$	−7.81966	−0.431112
$330$	0	0
$331$	2.79837	0.153813	0.0769063	−	0.997038i	$-0.475496\pi$
$331$	2.79837	0.153813	0.0769063	+	0.997038i	$0.475496\pi$
$332$	0	0
$333$	− 9.23607i	− 0.506133i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.0000i	0.653682i	0.945079	+	0.326841i	$0.105984\pi$
$337$	12.0000i	0.653682i	−0.945079	+	0.326841i	$0.894016\pi$
$338$	0	0
$339$	7.85410	0.426576
$340$	0	0
$341$	−8.36068	−0.452756
$342$	0	0
$343$	− 15.4164i	− 0.832408i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.7984i	− 0.901784i	−0.892579	−	0.450892i	$-0.851106\pi$
$347$	− 16.7984i	− 0.901784i	0.892579	−	0.450892i	$-0.148894\pi$
$348$	0	0
$349$	8.27051	0.442710	0.221355	−	0.975193i	$-0.428952\pi$
$349$	8.27051	0.442710	0.221355	+	0.975193i	$0.428952\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	22.2705i	1.18534i	0.805446	+	0.592670i	$0.201926\pi$
$353$	22.2705i	1.18534i	−0.805446	+	0.592670i	$0.798074\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.47214i	− 0.448393i
$358$	0	0
$359$	5.70820	0.301267	0.150634	−	0.988590i	$-0.451869\pi$
$359$	5.70820	0.301267	0.150634	+	0.988590i	$0.451869\pi$
$360$	0	0
$361$	55.2705	2.90897
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.2918i	1.26802i	0.773324	+	0.634011i	$0.218592\pi$
$367$	24.2918i	1.26802i	−0.773324	+	0.634011i	$0.781408\pi$
$368$	0	0
$369$	−8.94427	−0.465620
$370$	0	0
$371$	−1.34752	−0.0699600
$372$	0	0
$373$	− 14.6525i	− 0.758676i	−0.925258	−	0.379338i	$-0.876152\pi$
$373$	− 14.6525i	− 0.758676i	0.925258	−	0.379338i	$-0.123848\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.4721i	1.77541i
$378$	0	0
$379$	16.9443	0.870369	0.435184	−	0.900341i	$-0.356683\pi$
$379$	16.9443	0.870369	0.435184	+	0.900341i	$0.356683\pi$
$380$	0	0
$381$	18.4721	0.946356
$382$	0	0
$383$	3.90983i	0.199783i	0.994998	+	0.0998915i	$0.0318496\pi$
$383$	3.90983i	0.199783i	−0.994998	+	0.0998915i	$0.968150\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	−17.7082	−0.897842	−0.448921	−	0.893572i	$-0.648191\pi$
$389$	−17.7082	−0.897842	−0.448921	+	0.893572i	$0.648191\pi$
$390$	0	0
$391$	−16.3262	−0.825653
$392$	0	0
$393$	− 0.944272i	− 0.0476322i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.1803i	1.41433i	0.707048	+	0.707165i	$0.250026\pi$
$397$	28.1803i	1.41433i	−0.707048	+	0.707165i	$0.749974\pi$
$398$	0	0
$399$	−10.6525	−0.533291
$400$	0	0
$401$	−0.180340	−0.00900574	−0.00450287	−	0.999990i	$-0.501433\pi$
$401$	−0.180340	−0.00900574	−0.00450287	+	0.999990i	$0.501433\pi$
$402$	0	0
$403$	− 9.34752i	− 0.465633i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.9443i	1.83126i
$408$	0	0
$409$	−28.6869	−1.41848	−0.709238	−	0.704969i	$-0.750961\pi$
$409$	−28.6869	−1.41848	−0.709238	+	0.704969i	$0.750961\pi$
$410$	0	0
$411$	−10.9443	−0.539841
$412$	0	0
$413$	7.41641i	0.364938i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.32624i	− 0.456708i
$418$	0	0
$419$	24.7639	1.20980	0.604899	−	0.796302i	$-0.293214\pi$
$419$	24.7639	1.20980	0.604899	+	0.796302i	$0.293214\pi$
$420$	0	0
$421$	−2.43769	−0.118806	−0.0594030	−	0.998234i	$-0.518920\pi$
$421$	−2.43769	−0.118806	−0.0594030	+	0.998234i	$0.518920\pi$
$422$	0	0
$423$	6.32624i	0.307592i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.65248i	0.128362i
$428$	0	0
$429$	−17.8885	−0.863667
$430$	0	0
$431$	16.4721	0.793435	0.396717	−	0.917941i	$-0.370149\pi$
$431$	16.4721	0.793435	0.396717	+	0.917941i	$0.370149\pi$
$432$	0	0
$433$	− 3.23607i	− 0.155516i	−0.996972	−	0.0777578i	$-0.975224\pi$
$433$	− 3.23607i	− 0.155516i	0.996972	−	0.0777578i	$-0.0247761\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.5279i	0.981981i
$438$	0	0
$439$	−24.3607	−1.16267	−0.581336	−	0.813664i	$-0.697470\pi$
$439$	−24.3607	−1.16267	−0.581336	+	0.813664i	$0.697470\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	3.56231i	0.169250i	0.996413	+	0.0846251i	$0.0269693\pi$
$443$	3.56231i	0.169250i	−0.996413	+	0.0846251i	$0.973031\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 11.4164i	− 0.539978i
$448$	0	0
$449$	−34.8328	−1.64386	−0.821931	−	0.569587i	$-0.807103\pi$
$449$	−34.8328	−1.64386	−0.821931	+	0.569587i	$0.807103\pi$
$450$	0	0
$451$	35.7771	1.68468
$452$	0	0
$453$	5.85410i	0.275050i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.7639i	1.06485i	0.846477	+	0.532426i	$0.178720\pi$
$457$	22.7639i	1.06485i	−0.846477	+	0.532426i	$0.821280\pi$
$458$	0	0
$459$	−6.85410	−0.319922
$460$	0	0
$461$	3.88854	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$461$	3.88854	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$462$	0	0
$463$	− 0.763932i	− 0.0355029i	−0.999842	−	0.0177515i	$-0.994349\pi$
$463$	− 0.763932i	− 0.0355029i	0.999842	−	0.0177515i	$-0.00565076\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 21.6738i	− 1.00294i	−0.865174	−	0.501471i	$-0.832792\pi$
$467$	− 21.6738i	− 1.00294i	0.865174	−	0.501471i	$-0.167208\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−4.18034	−0.192620
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.09017i	0.0499155i
$478$	0	0
$479$	−37.7082	−1.72293	−0.861466	−	0.507815i	$-0.830453\pi$
$479$	−37.7082	−1.72293	−0.861466	+	0.507815i	$0.830453\pi$
$480$	0	0
$481$	−41.3050	−1.88334
$482$	0	0
$483$	− 2.94427i	− 0.133969i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.47214i	− 0.112023i	−0.998430	−	0.0560116i	$-0.982162\pi$
$487$	− 2.47214i	− 0.112023i	0.998430	−	0.0560116i	$-0.0178384\pi$
$488$	0	0
$489$	−8.29180	−0.374968
$490$	0	0
$491$	13.2361	0.597335	0.298668	−	0.954357i	$-0.403458\pi$
$491$	13.2361	0.597335	0.298668	+	0.954357i	$0.403458\pi$
$492$	0	0
$493$	− 52.8328i	− 2.37947i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.52786i	− 0.247959i
$498$	0	0
$499$	−30.7984	−1.37872	−0.689362	−	0.724417i	$-0.742109\pi$
$499$	−30.7984	−1.37872	−0.689362	+	0.724417i	$0.742109\pi$
$500$	0	0
$501$	−18.3262	−0.818756
$502$	0	0
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.00000i	− 0.310881i
$508$	0	0
$509$	29.7082	1.31679	0.658396	−	0.752671i	$-0.271235\pi$
$509$	29.7082	1.31679	0.658396	+	0.752671i	$0.271235\pi$
$510$	0	0
$511$	−16.9443	−0.749570
$512$	0	0
$513$	8.61803i	0.380495i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 25.3050i	− 1.11291i
$518$	0	0
$519$	−5.05573	−0.221922
$520$	0	0
$521$	15.8197	0.693072	0.346536	−	0.938037i	$-0.387358\pi$
$521$	15.8197	0.693072	0.346536	+	0.938037i	$0.387358\pi$
$522$	0	0
$523$	14.3607i	0.627949i	0.949431	+	0.313974i	$0.101661\pi$
$523$	14.3607i	0.627949i	−0.949431	+	0.313974i	$0.898339\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3262i	0.624061i
$528$	0	0
$529$	17.3262	0.753315
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	40.0000i	1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.23607i	0.0533403i
$538$	0	0
$539$	21.8885	0.942806
$540$	0	0
$541$	−9.43769	−0.405758	−0.202879	−	0.979204i	$-0.565030\pi$
$541$	−9.43769	−0.405758	−0.202879	+	0.979204i	$0.565030\pi$
$542$	0	0
$543$	11.3820i	0.488447i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.1803i	1.63247i	0.577718	+	0.816237i	$0.303944\pi$
$547$	38.1803i	1.63247i	−0.577718	+	0.816237i	$0.696056\pi$
$548$	0	0
$549$	2.14590	0.0915847
$550$	0	0
$551$	−66.4296	−2.82999
$552$	0	0
$553$	− 10.2918i	− 0.437652i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.11146i	− 0.174208i	−0.996199	−	0.0871040i	$-0.972239\pi$
$557$	− 4.11146i	− 0.174208i	0.996199	−	0.0871040i	$-0.0277612\pi$
$558$	0	0
$559$	−17.8885	−0.756605
$560$	0	0
$561$	27.4164	1.15752
$562$	0	0
$563$	29.7984i	1.25585i	0.778273	+	0.627926i	$0.216096\pi$
$563$	29.7984i	1.25585i	−0.778273	+	0.627926i	$0.783904\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.23607i	− 0.0519100i
$568$	0	0
$569$	33.5967	1.40845	0.704224	−	0.709977i	$-0.251295\pi$
$569$	33.5967	1.40845	0.704224	+	0.709977i	$0.251295\pi$
$570$	0	0
$571$	−23.4164	−0.979946	−0.489973	−	0.871738i	$-0.662993\pi$
$571$	−23.4164	−0.979946	−0.489973	+	0.871738i	$0.662993\pi$
$572$	0	0
$573$	21.1246i	0.882493i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.7639i	− 0.614631i	−0.951608	−	0.307315i	$-0.900569\pi$
$577$	− 14.7639i	− 0.614631i	0.951608	−	0.307315i	$-0.0994306\pi$
$578$	0	0
$579$	19.4164	0.806918
$580$	0	0
$581$	−4.40325	−0.182678
$582$	0	0
$583$	− 4.36068i	− 0.180601i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.49342i	0.185463i	0.995691	+	0.0927317i	$0.0295599\pi$
$587$	4.49342i	0.185463i	−0.995691	+	0.0927317i	$0.970440\pi$
$588$	0	0
$589$	18.0132	0.742219
$590$	0	0
$591$	22.0902	0.908668
$592$	0	0
$593$	− 37.7426i	− 1.54990i	−0.632020	−	0.774952i	$-0.717774\pi$
$593$	− 37.7426i	− 1.54990i	0.632020	−	0.774952i	$-0.282226\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.09017i	0.249254i
$598$	0	0
$599$	31.5279	1.28819	0.644097	−	0.764944i	$-0.277233\pi$
$599$	31.5279	1.28819	0.644097	+	0.764944i	$0.277233\pi$
$600$	0	0
$601$	−12.8541	−0.524330	−0.262165	−	0.965023i	$-0.584436\pi$
$601$	−12.8541	−0.524330	−0.262165	+	0.965023i	$0.584436\pi$
$602$	0	0
$603$	3.23607i	0.131783i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3.70820i	− 0.150511i	−0.997164	−	0.0752557i	$-0.976023\pi$
$607$	− 3.70820i	− 0.150511i	0.997164	−	0.0752557i	$-0.0239773\pi$
$608$	0	0
$609$	9.52786	0.386089
$610$	0	0
$611$	28.2918	1.14456
$612$	0	0
$613$	14.1803i	0.572739i	0.958119	+	0.286369i	$0.0924484\pi$
$613$	14.1803i	0.572739i	−0.958119	+	0.286369i	$0.907552\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.7426i	− 0.513000i	−0.966544	−	0.256500i	$-0.917431\pi$
$617$	− 12.7426i	− 0.513000i	0.966544	−	0.256500i	$-0.0825693\pi$
$618$	0	0
$619$	−5.14590	−0.206831	−0.103416	−	0.994638i	$-0.532977\pi$
$619$	−5.14590	−0.206831	−0.103416	+	0.994638i	$0.532977\pi$
$620$	0	0
$621$	−2.38197	−0.0955850
$622$	0	0
$623$	− 14.4721i	− 0.579814i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 34.4721i	− 1.37668i
$628$	0	0
$629$	63.3050	2.52413
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	0	0
$633$	14.2705i	0.567202i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	24.4721i	0.969621i
$638$	0	0
$639$	−4.47214	−0.176915
$640$	0	0
$641$	0.583592	0.0230505	0.0115253	−	0.999934i	$-0.496331\pi$
$641$	0.583592	0.0230505	0.0115253	+	0.999934i	$0.496331\pi$
$642$	0	0
$643$	− 15.0557i	− 0.593740i	−0.954918	−	0.296870i	$-0.904057\pi$
$643$	− 15.0557i	− 0.593740i	0.954918	−	0.296870i	$-0.0959428\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.58359i	− 0.337456i	−0.985663	−	0.168728i	$-0.946034\pi$
$647$	− 8.58359i	− 0.337456i	0.985663	−	0.168728i	$-0.0539659\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−2.58359	−0.101259
$652$	0	0
$653$	− 4.72949i	− 0.185079i	−0.995709	−	0.0925396i	$-0.970502\pi$
$653$	− 4.72949i	− 0.185079i	0.995709	−	0.0925396i	$-0.0294985\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.7082i	0.534808i
$658$	0	0
$659$	10.3607	0.403595	0.201797	−	0.979427i	$-0.435322\pi$
$659$	10.3607	0.403595	0.201797	+	0.979427i	$0.435322\pi$
$660$	0	0
$661$	−9.43769	−0.367084	−0.183542	−	0.983012i	$-0.558756\pi$
$661$	−9.43769	−0.367084	−0.183542	+	0.983012i	$0.558756\pi$
$662$	0	0
$663$	30.6525i	1.19044i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.3607i	− 0.710928i
$668$	0	0
$669$	7.23607	0.279763
$670$	0	0
$671$	−8.58359	−0.331366
$672$	0	0
$673$	28.9443i	1.11572i	0.829935	+	0.557860i	$0.188377\pi$
$673$	28.9443i	1.11572i	−0.829935	+	0.557860i	$0.811623\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.506578i	0.0194694i	0.999953	+	0.00973468i	$0.00309869\pi$
$677$	0.506578i	0.0194694i	−0.999953	+	0.00973468i	$0.996901\pi$
$678$	0	0
$679$	6.47214	0.248378
$680$	0	0
$681$	−9.79837	−0.375475
$682$	0	0
$683$	− 1.03444i	− 0.0395818i	−0.999804	−	0.0197909i	$-0.993700\pi$
$683$	− 1.03444i	− 0.0395818i	0.999804	−	0.0197909i	$-0.00630005\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.38197i	− 0.0908777i
$688$	0	0
$689$	4.87539	0.185737
$690$	0	0
$691$	15.6180	0.594138	0.297069	−	0.954856i	$-0.403991\pi$
$691$	15.6180	0.594138	0.297069	+	0.954856i	$0.403991\pi$
$692$	0	0
$693$	4.94427i	0.187817i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 61.3050i	− 2.32209i
$698$	0	0
$699$	−27.8885	−1.05484
$700$	0	0
$701$	34.5410	1.30460	0.652298	−	0.757962i	$-0.273805\pi$
$701$	34.5410	1.30460	0.652298	+	0.757962i	$0.273805\pi$
$702$	0	0
$703$	− 79.5967i	− 3.00205i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.360680i	0.0135648i
$708$	0	0
$709$	36.3262	1.36426	0.682130	−	0.731231i	$-0.261054\pi$
$709$	36.3262	1.36426	0.682130	+	0.731231i	$0.261054\pi$
$710$	0	0
$711$	−8.32624	−0.312258
$712$	0	0
$713$	4.97871i	0.186454i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.65248i	0.323133i
$718$	0	0
$719$	−8.18034	−0.305075	−0.152538	−	0.988298i	$-0.548745\pi$
$719$	−8.18034	−0.305075	−0.152538	+	0.988298i	$0.548745\pi$
$720$	0	0
$721$	−18.1115	−0.674506
$722$	0	0
$723$	14.3820i	0.534871i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.23607i	0.342547i	0.985224	+	0.171273i	$0.0547881\pi$
$727$	9.23607i	0.342547i	−0.985224	+	0.171273i	$0.945212\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	27.4164	1.01403
$732$	0	0
$733$	− 28.0689i	− 1.03675i	−0.855154	−	0.518374i	$-0.826538\pi$
$733$	− 28.0689i	− 1.03675i	0.855154	−	0.518374i	$-0.173462\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.9443i	− 0.476808i
$738$	0	0
$739$	5.32624	0.195929	0.0979644	−	0.995190i	$-0.468767\pi$
$739$	5.32624	0.195929	0.0979644	+	0.995190i	$0.468767\pi$
$740$	0	0
$741$	38.5410	1.41584
$742$	0	0
$743$	− 47.4164i	− 1.73954i	−0.493458	−	0.869770i	$-0.664267\pi$
$743$	− 47.4164i	− 1.73954i	0.493458	−	0.869770i	$-0.335733\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.56231i	0.130338i
$748$	0	0
$749$	13.0557	0.477046
$750$	0	0
$751$	7.67376	0.280020	0.140010	−	0.990150i	$-0.455287\pi$
$751$	7.67376	0.280020	0.140010	+	0.990150i	$0.455287\pi$
$752$	0	0
$753$	− 18.0000i	− 0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 28.4721i	− 1.03484i	−0.855732	−	0.517419i	$-0.826893\pi$
$757$	− 28.4721i	− 1.03484i	0.855732	−	0.517419i	$-0.173107\pi$
$758$	0	0
$759$	9.52786	0.345840
$760$	0	0
$761$	−53.7771	−1.94942	−0.974709	−	0.223478i	$-0.928259\pi$
$761$	−53.7771	−1.94942	−0.974709	+	0.223478i	$0.928259\pi$
$762$	0	0
$763$	− 20.1115i	− 0.728084i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 26.8328i	− 0.968877i
$768$	0	0
$769$	−3.38197	−0.121957	−0.0609784	−	0.998139i	$-0.519422\pi$
$769$	−3.38197	−0.121957	−0.0609784	+	0.998139i	$0.519422\pi$
$770$	0	0
$771$	3.20163	0.115304
$772$	0	0
$773$	− 10.0344i	− 0.360914i	−0.983583	−	0.180457i	$-0.942242\pi$
$773$	− 10.0344i	− 0.360914i	0.983583	−	0.180457i	$-0.0577576\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.4164i	0.409561i
$778$	0	0
$779$	−77.0820	−2.76175
$780$	0	0
$781$	17.8885	0.640102
$782$	0	0
$783$	− 7.70820i	− 0.275469i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.7082i	1.20157i	0.799412	+	0.600784i	$0.205145\pi$
$787$	33.7082i	1.20157i	−0.799412	+	0.600784i	$0.794855\pi$
$788$	0	0
$789$	24.0902	0.857633
$790$	0	0
$791$	−9.70820	−0.345184
$792$	0	0
$793$	− 9.59675i	− 0.340791i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	51.2148i	1.81412i	0.421001	+	0.907060i	$0.361679\pi$
$797$	51.2148i	1.81412i	−0.421001	+	0.907060i	$0.638321\pi$
$798$	0	0
$799$	−43.3607	−1.53399
$800$	0	0
$801$	−11.7082	−0.413689
$802$	0	0
$803$	− 54.8328i	− 1.93501i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.1246i	0.673218i
$808$	0	0
$809$	−28.7639	−1.01129	−0.505643	−	0.862743i	$-0.668745\pi$
$809$	−28.7639	−1.01129	−0.505643	+	0.862743i	$0.668745\pi$
$810$	0	0
$811$	−36.1033	−1.26776	−0.633880	−	0.773432i	$-0.718539\pi$
$811$	−36.1033	−1.26776	−0.633880	+	0.773432i	$0.718539\pi$
$812$	0	0
$813$	28.5623i	1.00172i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 34.4721i	− 1.20603i
$818$	0	0
$819$	−5.52786	−0.193159
$820$	0	0
$821$	−1.63932	−0.0572127	−0.0286063	−	0.999591i	$-0.509107\pi$
$821$	−1.63932	−0.0572127	−0.0286063	+	0.999591i	$0.509107\pi$
$822$	0	0
$823$	44.7639i	1.56037i	0.625547	+	0.780186i	$0.284876\pi$
$823$	44.7639i	1.56037i	−0.625547	+	0.780186i	$0.715124\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 21.0902i	− 0.733377i	−0.930344	−	0.366689i	$-0.880491\pi$
$827$	− 21.0902i	− 0.733377i	0.930344	−	0.366689i	$-0.119509\pi$
$828$	0	0
$829$	7.67376	0.266521	0.133260	−	0.991081i	$-0.457455\pi$
$829$	7.67376	0.266521	0.133260	+	0.991081i	$0.457455\pi$
$830$	0	0
$831$	−10.7639	−0.373397
$832$	0	0
$833$	− 37.5066i	− 1.29953i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.09017i	0.0722468i
$838$	0	0
$839$	−36.6525	−1.26538	−0.632692	−	0.774404i	$-0.718050\pi$
$839$	−36.6525	−1.26538	−0.632692	+	0.774404i	$0.718050\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	3.23607i	0.111456i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.18034i	− 0.212359i
$848$	0	0
$849$	32.0689	1.10060
$850$	0	0
$851$	22.0000	0.754150
$852$	0	0
$853$	− 49.9574i	− 1.71051i	−0.518208	−	0.855255i	$-0.673401\pi$
$853$	− 49.9574i	− 1.71051i	0.518208	−	0.855255i	$-0.326599\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.8115i	1.56489i	0.622718	+	0.782446i	$0.286028\pi$
$857$	45.8115i	1.56489i	−0.622718	+	0.782446i	$0.713972\pi$
$858$	0	0
$859$	−17.8885	−0.610349	−0.305175	−	0.952296i	$-0.598715\pi$
$859$	−17.8885	−0.610349	−0.305175	+	0.952296i	$0.598715\pi$
$860$	0	0
$861$	11.0557	0.376778
$862$	0	0
$863$	45.8885i	1.56206i	0.624491	+	0.781032i	$0.285307\pi$
$863$	45.8885i	1.56206i	−0.624491	+	0.781032i	$0.714693\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 29.9787i	− 1.01813i
$868$	0	0
$869$	33.3050	1.12979
$870$	0	0
$871$	14.4721	0.490370
$872$	0	0
$873$	− 5.23607i	− 0.177214i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 24.2492i	− 0.818838i	−0.912346	−	0.409419i	$-0.865731\pi$
$877$	− 24.2492i	− 0.818838i	0.912346	−	0.409419i	$-0.134269\pi$
$878$	0	0
$879$	17.3262	0.584400
$880$	0	0
$881$	4.06888	0.137084	0.0685421	−	0.997648i	$-0.478165\pi$
$881$	4.06888	0.137084	0.0685421	+	0.997648i	$0.478165\pi$
$882$	0	0
$883$	− 20.2918i	− 0.682873i	−0.939905	−	0.341437i	$-0.889087\pi$
$883$	− 20.2918i	− 0.682873i	0.939905	−	0.341437i	$-0.110913\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.79837i	− 0.295420i	−0.989031	−	0.147710i	$-0.952810\pi$
$887$	− 8.79837i	− 0.295420i	0.989031	−	0.147710i	$-0.0471903\pi$
$888$	0	0
$889$	−22.8328	−0.765788
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	54.5197i	1.82443i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.6525i	0.355676i
$898$	0	0
$899$	−16.1115	−0.537347
$900$	0	0
$901$	−7.47214	−0.248933
$902$	0	0
$903$	4.94427i	0.164535i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.7771i	1.12155i	0.827968	+	0.560775i	$0.189497\pi$
$907$	33.7771i	1.12155i	−0.827968	+	0.560775i	$0.810503\pi$
$908$	0	0
$909$	0.291796	0.00967826
$910$	0	0
$911$	−6.94427	−0.230074	−0.115037	−	0.993361i	$-0.536699\pi$
$911$	−6.94427	−0.230074	−0.115037	+	0.993361i	$0.536699\pi$
$912$	0	0
$913$	− 14.2492i	− 0.471580i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.16718i	0.0385438i
$918$	0	0
$919$	−26.6738	−0.879886	−0.439943	−	0.898026i	$-0.645001\pi$
$919$	−26.6738	−0.879886	−0.439943	+	0.898026i	$0.645001\pi$
$920$	0	0
$921$	−32.4721	−1.06999
$922$	0	0
$923$	20.0000i	0.658308i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.6525i	0.481250i
$928$	0	0
$929$	−56.5410	−1.85505	−0.927525	−	0.373760i	$-0.878068\pi$
$929$	−56.5410	−1.85505	−0.927525	+	0.373760i	$0.878068\pi$
$930$	0	0
$931$	−47.1591	−1.54558
$932$	0	0
$933$	32.1803i	1.05354i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 14.1803i	− 0.463252i	−0.972805	−	0.231626i	$-0.925596\pi$
$937$	− 14.1803i	− 0.463252i	0.972805	−	0.231626i	$-0.0744045\pi$
$938$	0	0
$939$	2.58359	0.0843123
$940$	0	0
$941$	−41.3050	−1.34650	−0.673251	−	0.739414i	$-0.735103\pi$
$941$	−41.3050	−1.34650	−0.673251	+	0.739414i	$0.735103\pi$
$942$	0	0
$943$	− 21.3050i	− 0.693785i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 57.5755i	− 1.87095i	−0.353391	−	0.935476i	$-0.614972\pi$
$947$	− 57.5755i	− 1.87095i	0.353391	−	0.935476i	$-0.385028\pi$
$948$	0	0
$949$	61.3050	1.99004
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	− 33.6869i	− 1.09123i	−0.838037	−	0.545613i	$-0.816297\pi$
$953$	− 33.6869i	− 1.09123i	0.838037	−	0.545613i	$-0.183703\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	30.8328i	0.996683i
$958$	0	0
$959$	13.5279	0.436838
$960$	0	0
$961$	−26.6312	−0.859071
$962$	0	0
$963$	− 10.5623i	− 0.340366i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 21.7082i	− 0.698089i	−0.937106	−	0.349044i	$-0.886506\pi$
$967$	− 21.7082i	− 0.698089i	0.937106	−	0.349044i	$-0.113494\pi$
$968$	0	0
$969$	−59.0689	−1.89757
$970$	0	0
$971$	−34.8328	−1.11784	−0.558919	−	0.829222i	$-0.688784\pi$
$971$	−34.8328	−1.11784	−0.558919	+	0.829222i	$0.688784\pi$
$972$	0	0
$973$	11.5279i	0.369566i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28.3820i	− 0.908020i	−0.890997	−	0.454010i	$-0.849993\pi$
$977$	− 28.3820i	− 0.908020i	0.890997	−	0.454010i	$-0.150007\pi$
$978$	0	0
$979$	46.8328	1.49678
$980$	0	0
$981$	−16.2705	−0.519477
$982$	0	0
$983$	18.9098i	0.603130i	0.953446	+	0.301565i	$0.0975090\pi$
$983$	18.9098i	0.603130i	−0.953446	+	0.301565i	$0.902491\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 7.81966i	− 0.248903i
$988$	0	0
$989$	9.52786	0.302968
$990$	0	0
$991$	15.7426	0.500082	0.250041	−	0.968235i	$-0.419556\pi$
$991$	15.7426	0.500082	0.250041	+	0.968235i	$0.419556\pi$
$992$	0	0
$993$	2.79837i	0.0888037i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.7214i	− 1.22632i	−0.789960	−	0.613159i	$-0.789899\pi$
$997$	− 38.7214i	− 1.22632i	0.789960	−	0.613159i	$-0.210101\pi$
$998$	0	0
$999$	9.23607	0.292216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.m.1249.3		4
4.3	odd	2		1500.2.d.a.1249.2		4
5.2	odd	4		6000.2.a.t.1.2		2
5.3	odd	4		6000.2.a.j.1.1		2
5.4	even	2	inner	6000.2.f.m.1249.2		4
12.11	even	2		4500.2.d.j.4249.3		4
20.3	even	4		1500.2.a.f.1.2	yes	2
20.7	even	4		1500.2.a.b.1.1	✓	2
20.19	odd	2		1500.2.d.a.1249.3		4
60.23	odd	4		4500.2.a.g.1.2		2
60.47	odd	4		4500.2.a.k.1.1		2
60.59	even	2		4500.2.d.j.4249.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.b.1.1	✓	2	20.7	even	4
1500.2.a.f.1.2	yes	2	20.3	even	4
1500.2.d.a.1249.2		4	4.3	odd	2
1500.2.d.a.1249.3		4	20.19	odd	2
4500.2.a.g.1.2		2	60.23	odd	4
4500.2.a.k.1.1		2	60.47	odd	4
4500.2.d.j.4249.2		4	60.59	even	2
4500.2.d.j.4249.3		4	12.11	even	2
6000.2.a.j.1.1		2	5.3	odd	4
6000.2.a.t.1.2		2	5.2	odd	4
6000.2.f.m.1249.2		4	5.4	even	2	inner
6000.2.f.m.1249.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.23607i q^{7} -1.00000 q^{9} +4.00000 q^{11} +4.47214i q^{13} -6.85410i q^{17} -8.61803 q^{19} +1.23607 q^{21} -2.38197i q^{23} -1.00000i q^{27} +7.70820 q^{29} -2.09017 q^{31} +4.00000i q^{33} +9.23607i q^{37} -4.47214 q^{39} +8.94427 q^{41} +4.00000i q^{43} -6.32624i q^{47} +5.47214 q^{49} +6.85410 q^{51} -1.09017i q^{53} -8.61803i q^{57} -6.00000 q^{59} -2.14590 q^{61} +1.23607i q^{63} -3.23607i q^{67} +2.38197 q^{69} +4.47214 q^{71} -13.7082i q^{73} -4.94427i q^{77} +8.32624 q^{79} +1.00000 q^{81} -3.56231i q^{83} +7.70820i q^{87} +11.7082 q^{89} +5.52786 q^{91} -2.09017i q^{93} +5.23607i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 16 q^{11} - 30 q^{19} - 4 q^{21} + 4 q^{29} + 14 q^{31} + 4 q^{49} + 14 q^{51} - 24 q^{59} - 22 q^{61} + 14 q^{69} + 2 q^{79} + 4 q^{81} + 20 q^{89} + 40 q^{91} - 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 16 * q^11 - 30 * q^19 - 4 * q^21 + 4 * q^29 + 14 * q^31 + 4 * q^49 + 14 * q^51 - 24 * q^59 - 22 * q^61 + 14 * q^69 + 2 * q^79 + 4 * q^81 + 20 * q^89 + 40 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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