LMFDB - Embedded newform 1200.4.f.m.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,4,Mod(49,1200)] 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("1200.49"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-18,0,48,0,0,0,0,0,0,0,-248] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.m.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +20.0000i q^{7} -9.00000 q^{9} +24.0000 q^{11} +74.0000i q^{13} -54.0000i q^{17} -124.000 q^{19} -60.0000 q^{21} +120.000i q^{23} -27.0000i q^{27} +78.0000 q^{29} -200.000 q^{31} +72.0000i q^{33} +70.0000i q^{37} -222.000 q^{39} +330.000 q^{41} -92.0000i q^{43} -24.0000i q^{47} -57.0000 q^{49} +162.000 q^{51} +450.000i q^{53} -372.000i q^{57} +24.0000 q^{59} -322.000 q^{61} -180.000i q^{63} -196.000i q^{67} -360.000 q^{69} +288.000 q^{71} -430.000i q^{73} +480.000i q^{77} -520.000 q^{79} +81.0000 q^{81} -156.000i q^{83} +234.000i q^{87} -1026.00 q^{89} -1480.00 q^{91} -600.000i q^{93} +286.000i q^{97} -216.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 48 q^{11} - 248 q^{19} - 120 q^{21} + 156 q^{29} - 400 q^{31} - 444 q^{39} + 660 q^{41} - 114 q^{49} + 324 q^{51} + 48 q^{59} - 644 q^{61} - 720 q^{69} + 576 q^{71} - 1040 q^{79} + 162 q^{81}+ \cdots - 432 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 48 * q^11 - 248 * q^19 - 120 * q^21 + 156 * q^29 - 400 * q^31 - 444 * q^39 + 660 * q^41 - 114 * q^49 + 324 * q^51 + 48 * q^59 - 644 * q^61 - 720 * q^69 + 576 * q^71 - 1040 * q^79 + 162 * q^81 - 2052 * q^89 - 2960 * q^91 - 432 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.0000i	1.07990i	0.841698	+	0.539949i	$0.181557\pi$
$7$	20.0000i	1.07990i	−0.841698	+	0.539949i	$0.818443\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	74.0000i	1.57876i	0.613904	+	0.789381i	$0.289598\pi$
$13$	74.0000i	1.57876i	−0.613904	+	0.789381i	$0.710402\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 54.0000i	− 0.770407i	−0.922832	−	0.385204i	$-0.874131\pi$
$17$	− 54.0000i	− 0.770407i	0.922832	−	0.385204i	$-0.125869\pi$
$18$	0	0
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	0	0
$21$	−60.0000	−0.623480
$22$	0	0
$23$	120.000i	1.08790i	0.839117	+	0.543951i	$0.183072\pi$
$23$	120.000i	1.08790i	−0.839117	+	0.543951i	$0.816928\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	78.0000	0.499456	0.249728	−	0.968316i	$-0.419659\pi$
$29$	78.0000	0.499456	0.249728	+	0.968316i	$0.419659\pi$
$30$	0	0
$31$	−200.000	−1.15874	−0.579372	−	0.815063i	$-0.696702\pi$
$31$	−200.000	−1.15874	−0.579372	+	0.815063i	$0.696702\pi$
$32$	0	0
$33$	72.0000i	0.379806i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	70.0000i	0.311025i	0.987834	+	0.155513i	$0.0497029\pi$
$37$	70.0000i	0.311025i	−0.987834	+	0.155513i	$0.950297\pi$
$38$	0	0
$39$	−222.000	−0.911499
$40$	0	0
$41$	330.000	1.25701	0.628504	−	0.777806i	$-0.283668\pi$
$41$	330.000	1.25701	0.628504	+	0.777806i	$0.283668\pi$
$42$	0	0
$43$	− 92.0000i	− 0.326276i	−0.986603	−	0.163138i	$-0.947838\pi$
$43$	− 92.0000i	− 0.326276i	0.986603	−	0.163138i	$-0.0521616\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 24.0000i	− 0.0744843i	−0.999306	−	0.0372421i	$-0.988143\pi$
$47$	− 24.0000i	− 0.0744843i	0.999306	−	0.0372421i	$-0.0118573\pi$
$48$	0	0
$49$	−57.0000	−0.166181
$50$	0	0
$51$	162.000	0.444795
$52$	0	0
$53$	450.000i	1.16627i	0.812376	+	0.583134i	$0.198174\pi$
$53$	450.000i	1.16627i	−0.812376	+	0.583134i	$0.801826\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 372.000i	− 0.864432i
$58$	0	0
$59$	24.0000	0.0529582	0.0264791	−	0.999649i	$-0.491570\pi$
$59$	24.0000	0.0529582	0.0264791	+	0.999649i	$0.491570\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	0	0
$63$	− 180.000i	− 0.359966i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 196.000i	− 0.357391i	−0.983904	−	0.178696i	$-0.942812\pi$
$67$	− 196.000i	− 0.357391i	0.983904	−	0.178696i	$-0.0571877\pi$
$68$	0	0
$69$	−360.000	−0.628100
$70$	0	0
$71$	288.000	0.481399	0.240699	−	0.970600i	$-0.422623\pi$
$71$	288.000	0.481399	0.240699	+	0.970600i	$0.422623\pi$
$72$	0	0
$73$	− 430.000i	− 0.689420i	−0.938709	−	0.344710i	$-0.887977\pi$
$73$	− 430.000i	− 0.689420i	0.938709	−	0.344710i	$-0.112023\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	480.000i	0.710404i
$78$	0	0
$79$	−520.000	−0.740564	−0.370282	−	0.928919i	$-0.620739\pi$
$79$	−520.000	−0.740564	−0.370282	+	0.928919i	$0.620739\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 156.000i	− 0.206304i	−0.994666	−	0.103152i	$-0.967107\pi$
$83$	− 156.000i	− 0.206304i	0.994666	−	0.103152i	$-0.0328928\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	234.000i	0.288361i
$88$	0	0
$89$	−1026.00	−1.22198	−0.610988	−	0.791640i	$-0.709227\pi$
$89$	−1026.00	−1.22198	−0.610988	+	0.791640i	$0.709227\pi$
$90$	0	0
$91$	−1480.00	−1.70490
$92$	0	0
$93$	− 600.000i	− 0.669001i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	286.000i	0.299370i	0.988734	+	0.149685i	$0.0478260\pi$
$97$	286.000i	0.299370i	−0.988734	+	0.149685i	$0.952174\pi$
$98$	0	0
$99$	−216.000	−0.219281
$100$	0	0
$101$	−1734.00	−1.70831	−0.854156	−	0.520017i	$-0.825925\pi$
$101$	−1734.00	−1.70831	−0.854156	+	0.520017i	$0.825925\pi$
$102$	0	0
$103$	− 452.000i	− 0.432397i	−0.976349	−	0.216198i	$-0.930634\pi$
$103$	− 452.000i	− 0.432397i	0.976349	−	0.216198i	$-0.0693658\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1404.00i	− 1.26850i	−0.773127	−	0.634251i	$-0.781308\pi$
$107$	− 1404.00i	− 1.26850i	0.773127	−	0.634251i	$-0.218692\pi$
$108$	0	0
$109$	1474.00	1.29526	0.647631	−	0.761954i	$-0.275760\pi$
$109$	1474.00	1.29526	0.647631	+	0.761954i	$0.275760\pi$
$110$	0	0
$111$	−210.000	−0.179570
$112$	0	0
$113$	1086.00i	0.904091i	0.891995	+	0.452046i	$0.149306\pi$
$113$	1086.00i	0.904091i	−0.891995	+	0.452046i	$0.850694\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 666.000i	− 0.526254i
$118$	0	0
$119$	1080.00	0.831962
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	990.000i	0.725734i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1244.00i	0.869190i	0.900626	+	0.434595i	$0.143109\pi$
$127$	1244.00i	0.869190i	−0.900626	+	0.434595i	$0.856891\pi$
$128$	0	0
$129$	276.000	0.188376
$130$	0	0
$131$	−2328.00	−1.55266	−0.776329	−	0.630327i	$-0.782921\pi$
$131$	−2328.00	−1.55266	−0.776329	+	0.630327i	$0.782921\pi$
$132$	0	0
$133$	− 2480.00i	− 1.61687i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2118.00i	− 1.32082i	−0.750903	−	0.660412i	$-0.770382\pi$
$137$	− 2118.00i	− 1.32082i	0.750903	−	0.660412i	$-0.229618\pi$
$138$	0	0
$139$	2324.00	1.41812	0.709062	−	0.705147i	$-0.249119\pi$
$139$	2324.00	1.41812	0.709062	+	0.705147i	$0.249119\pi$
$140$	0	0
$141$	72.0000	0.0430035
$142$	0	0
$143$	1776.00i	1.03858i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 171.000i	− 0.0959445i
$148$	0	0
$149$	−258.000	−0.141854	−0.0709268	−	0.997482i	$-0.522596\pi$
$149$	−258.000	−0.141854	−0.0709268	+	0.997482i	$0.522596\pi$
$150$	0	0
$151$	808.000	0.435458	0.217729	−	0.976009i	$-0.430135\pi$
$151$	808.000	0.435458	0.217729	+	0.976009i	$0.430135\pi$
$152$	0	0
$153$	486.000i	0.256802i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2378.00i	− 1.20882i	−0.796673	−	0.604411i	$-0.793408\pi$
$157$	− 2378.00i	− 1.20882i	0.796673	−	0.604411i	$-0.206592\pi$
$158$	0	0
$159$	−1350.00	−0.673346
$160$	0	0
$161$	−2400.00	−1.17482
$162$	0	0
$163$	52.0000i	0.0249874i	0.999922	+	0.0124937i	$0.00397698\pi$
$163$	52.0000i	0.0249874i	−0.999922	+	0.0124937i	$0.996023\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3720.00i	− 1.72373i	−0.507141	−	0.861863i	$-0.669298\pi$
$167$	− 3720.00i	− 1.72373i	0.507141	−	0.861863i	$-0.330702\pi$
$168$	0	0
$169$	−3279.00	−1.49249
$170$	0	0
$171$	1116.00	0.499080
$172$	0	0
$173$	426.000i	0.187215i	0.995609	+	0.0936075i	$0.0298399\pi$
$173$	426.000i	0.187215i	−0.995609	+	0.0936075i	$0.970160\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	72.0000i	0.0305754i
$178$	0	0
$179$	−1440.00	−0.601289	−0.300644	−	0.953736i	$-0.597202\pi$
$179$	−1440.00	−0.601289	−0.300644	+	0.953736i	$0.597202\pi$
$180$	0	0
$181$	−3130.00	−1.28537	−0.642683	−	0.766133i	$-0.722179\pi$
$181$	−3130.00	−1.28537	−0.642683	+	0.766133i	$0.722179\pi$
$182$	0	0
$183$	− 966.000i	− 0.390212i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1296.00i	− 0.506807i
$188$	0	0
$189$	540.000	0.207827
$190$	0	0
$191$	−3576.00	−1.35471	−0.677357	−	0.735655i	$-0.736875\pi$
$191$	−3576.00	−1.35471	−0.677357	+	0.735655i	$0.736875\pi$
$192$	0	0
$193$	2666.00i	0.994315i	0.867660	+	0.497158i	$0.165623\pi$
$193$	2666.00i	0.994315i	−0.867660	+	0.497158i	$0.834377\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2718.00i	0.982992i	0.870880	+	0.491496i	$0.163550\pi$
$197$	2718.00i	0.982992i	−0.870880	+	0.491496i	$0.836450\pi$
$198$	0	0
$199$	−3832.00	−1.36504	−0.682521	−	0.730866i	$-0.739116\pi$
$199$	−3832.00	−1.36504	−0.682521	+	0.730866i	$0.739116\pi$
$200$	0	0
$201$	588.000	0.206340
$202$	0	0
$203$	1560.00i	0.539362i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1080.00i	− 0.362634i
$208$	0	0
$209$	−2976.00	−0.984948
$210$	0	0
$211$	−1100.00	−0.358896	−0.179448	−	0.983767i	$-0.557431\pi$
$211$	−1100.00	−0.358896	−0.179448	+	0.983767i	$0.557431\pi$
$212$	0	0
$213$	864.000i	0.277936i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4000.00i	− 1.25133i
$218$	0	0
$219$	1290.00	0.398037
$220$	0	0
$221$	3996.00	1.21629
$222$	0	0
$223$	− 1964.00i	− 0.589772i	−0.955532	−	0.294886i	$-0.904718\pi$
$223$	− 1964.00i	− 0.589772i	0.955532	−	0.294886i	$-0.0952817\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	660.000i	0.192977i	0.995334	+	0.0964884i	$0.0307611\pi$
$227$	660.000i	0.192977i	−0.995334	+	0.0964884i	$0.969239\pi$
$228$	0	0
$229$	1906.00	0.550009	0.275004	−	0.961443i	$-0.411321\pi$
$229$	1906.00	0.550009	0.275004	+	0.961443i	$0.411321\pi$
$230$	0	0
$231$	−1440.00	−0.410152
$232$	0	0
$233$	− 1458.00i	− 0.409943i	−0.978768	−	0.204972i	$-0.934290\pi$
$233$	− 1458.00i	− 0.409943i	0.978768	−	0.204972i	$-0.0657102\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1560.00i	− 0.427565i
$238$	0	0
$239$	1176.00	0.318281	0.159140	−	0.987256i	$-0.449128\pi$
$239$	1176.00	0.318281	0.159140	+	0.987256i	$0.449128\pi$
$240$	0	0
$241$	866.000	0.231469	0.115734	−	0.993280i	$-0.463078\pi$
$241$	866.000	0.231469	0.115734	+	0.993280i	$0.463078\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9176.00i	− 2.36379i
$248$	0	0
$249$	468.000	0.119110
$250$	0	0
$251$	−432.000	−0.108636	−0.0543179	−	0.998524i	$-0.517298\pi$
$251$	−432.000	−0.108636	−0.0543179	+	0.998524i	$0.517298\pi$
$252$	0	0
$253$	2880.00i	0.715668i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2526.00i	− 0.613103i	−0.951854	−	0.306552i	$-0.900825\pi$
$257$	− 2526.00i	− 0.613103i	0.951854	−	0.306552i	$-0.0991752\pi$
$258$	0	0
$259$	−1400.00	−0.335876
$260$	0	0
$261$	−702.000	−0.166485
$262$	0	0
$263$	− 5448.00i	− 1.27733i	−0.769484	−	0.638666i	$-0.779487\pi$
$263$	− 5448.00i	− 1.27733i	0.769484	−	0.638666i	$-0.220513\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3078.00i	− 0.705508i
$268$	0	0
$269$	2574.00	0.583418	0.291709	−	0.956507i	$-0.405776\pi$
$269$	2574.00	0.583418	0.291709	+	0.956507i	$0.405776\pi$
$270$	0	0
$271$	3184.00	0.713706	0.356853	−	0.934161i	$-0.383850\pi$
$271$	3184.00	0.713706	0.356853	+	0.934161i	$0.383850\pi$
$272$	0	0
$273$	− 4440.00i	− 0.984326i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 3962.00i	− 0.859399i	−0.902972	−	0.429699i	$-0.858620\pi$
$277$	− 3962.00i	− 0.859399i	0.902972	−	0.429699i	$-0.141380\pi$
$278$	0	0
$279$	1800.00	0.386248
$280$	0	0
$281$	−8286.00	−1.75908	−0.879540	−	0.475825i	$-0.842149\pi$
$281$	−8286.00	−1.75908	−0.879540	+	0.475825i	$0.842149\pi$
$282$	0	0
$283$	2716.00i	0.570493i	0.958454	+	0.285246i	$0.0920754\pi$
$283$	2716.00i	0.570493i	−0.958454	+	0.285246i	$0.907925\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6600.00i	1.35744i
$288$	0	0
$289$	1997.00	0.406473
$290$	0	0
$291$	−858.000	−0.172841
$292$	0	0
$293$	6018.00i	1.19992i	0.800032	+	0.599958i	$0.204816\pi$
$293$	6018.00i	1.19992i	−0.800032	+	0.599958i	$0.795184\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 648.000i	− 0.126602i
$298$	0	0
$299$	−8880.00	−1.71754
$300$	0	0
$301$	1840.00	0.352345
$302$	0	0
$303$	− 5202.00i	− 0.986294i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9236.00i	1.71702i	0.512793	+	0.858512i	$0.328611\pi$
$307$	9236.00i	1.71702i	−0.512793	+	0.858512i	$0.671389\pi$
$308$	0	0
$309$	1356.00	0.249644
$310$	0	0
$311$	−1536.00	−0.280060	−0.140030	−	0.990147i	$-0.544720\pi$
$311$	−1536.00	−0.280060	−0.140030	+	0.990147i	$0.544720\pi$
$312$	0	0
$313$	− 7342.00i	− 1.32586i	−0.748681	−	0.662930i	$-0.769313\pi$
$313$	− 7342.00i	− 1.32586i	0.748681	−	0.662930i	$-0.230687\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3894.00i	0.689933i	0.938615	+	0.344967i	$0.112110\pi$
$317$	3894.00i	0.689933i	−0.938615	+	0.344967i	$0.887890\pi$
$318$	0	0
$319$	1872.00	0.328564
$320$	0	0
$321$	4212.00	0.732370
$322$	0	0
$323$	6696.00i	1.15348i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4422.00i	0.747820i
$328$	0	0
$329$	480.000	0.0804354
$330$	0	0
$331$	−3692.00	−0.613084	−0.306542	−	0.951857i	$-0.599172\pi$
$331$	−3692.00	−0.613084	−0.306542	+	0.951857i	$0.599172\pi$
$332$	0	0
$333$	− 630.000i	− 0.103675i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8998.00i	1.45446i	0.686395	+	0.727229i	$0.259192\pi$
$337$	8998.00i	1.45446i	−0.686395	+	0.727229i	$0.740808\pi$
$338$	0	0
$339$	−3258.00	−0.521977
$340$	0	0
$341$	−4800.00	−0.762271
$342$	0	0
$343$	5720.00i	0.900440i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5244.00i	0.811276i	0.914034	+	0.405638i	$0.132951\pi$
$347$	5244.00i	0.811276i	−0.914034	+	0.405638i	$0.867049\pi$
$348$	0	0
$349$	−6302.00	−0.966585	−0.483293	−	0.875459i	$-0.660559\pi$
$349$	−6302.00	−0.966585	−0.483293	+	0.875459i	$0.660559\pi$
$350$	0	0
$351$	1998.00	0.303833
$352$	0	0
$353$	3414.00i	0.514756i	0.966311	+	0.257378i	$0.0828586\pi$
$353$	3414.00i	0.514756i	−0.966311	+	0.257378i	$0.917141\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3240.00i	0.480333i
$358$	0	0
$359$	4824.00	0.709195	0.354597	−	0.935019i	$-0.384618\pi$
$359$	4824.00	0.709195	0.354597	+	0.935019i	$0.384618\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	0	0
$363$	− 2265.00i	− 0.327498i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3508.00i	− 0.498954i	−0.968381	−	0.249477i	$-0.919741\pi$
$367$	− 3508.00i	− 0.498954i	0.968381	−	0.249477i	$-0.0802587\pi$
$368$	0	0
$369$	−2970.00	−0.419003
$370$	0	0
$371$	−9000.00	−1.25945
$372$	0	0
$373$	10802.0i	1.49948i	0.661732	+	0.749740i	$0.269822\pi$
$373$	10802.0i	1.49948i	−0.661732	+	0.749740i	$0.730178\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5772.00i	0.788523i
$378$	0	0
$379$	1460.00	0.197876	0.0989382	−	0.995094i	$-0.468455\pi$
$379$	1460.00	0.197876	0.0989382	+	0.995094i	$0.468455\pi$
$380$	0	0
$381$	−3732.00	−0.501827
$382$	0	0
$383$	4872.00i	0.649994i	0.945715	+	0.324997i	$0.105363\pi$
$383$	4872.00i	0.649994i	−0.945715	+	0.324997i	$0.894637\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	828.000i	0.108759i
$388$	0	0
$389$	14046.0	1.83075	0.915373	−	0.402606i	$-0.131896\pi$
$389$	14046.0	1.83075	0.915373	+	0.402606i	$0.131896\pi$
$390$	0	0
$391$	6480.00	0.838127
$392$	0	0
$393$	− 6984.00i	− 0.896428i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2734.00i	0.345631i	0.984954	+	0.172816i	$0.0552864\pi$
$397$	2734.00i	0.345631i	−0.984954	+	0.172816i	$0.944714\pi$
$398$	0	0
$399$	7440.00	0.933498
$400$	0	0
$401$	−15942.0	−1.98530	−0.992650	−	0.121019i	$-0.961384\pi$
$401$	−15942.0	−1.98530	−0.992650	+	0.121019i	$0.961384\pi$
$402$	0	0
$403$	− 14800.0i	− 1.82938i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1680.00i	0.204606i
$408$	0	0
$409$	−8714.00	−1.05350	−0.526748	−	0.850022i	$-0.676589\pi$
$409$	−8714.00	−1.05350	−0.526748	+	0.850022i	$0.676589\pi$
$410$	0	0
$411$	6354.00	0.762578
$412$	0	0
$413$	480.000i	0.0571895i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6972.00i	0.818754i
$418$	0	0
$419$	11976.0	1.39634	0.698169	−	0.715933i	$-0.253998\pi$
$419$	11976.0	1.39634	0.698169	+	0.715933i	$0.253998\pi$
$420$	0	0
$421$	11054.0	1.27967	0.639833	−	0.768514i	$-0.279004\pi$
$421$	11054.0	1.27967	0.639833	+	0.768514i	$0.279004\pi$
$422$	0	0
$423$	216.000i	0.0248281i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6440.00i	− 0.729868i
$428$	0	0
$429$	−5328.00	−0.599623
$430$	0	0
$431$	−720.000	−0.0804668	−0.0402334	−	0.999190i	$-0.512810\pi$
$431$	−720.000	−0.0804668	−0.0402334	+	0.999190i	$0.512810\pi$
$432$	0	0
$433$	− 15622.0i	− 1.73382i	−0.498462	−	0.866912i	$-0.666102\pi$
$433$	− 15622.0i	− 1.73382i	0.498462	−	0.866912i	$-0.333898\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 14880.0i	− 1.62885i
$438$	0	0
$439$	−9880.00	−1.07414	−0.537069	−	0.843538i	$-0.680469\pi$
$439$	−9880.00	−1.07414	−0.537069	+	0.843538i	$0.680469\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	16116.0i	1.72843i	0.503123	+	0.864215i	$0.332184\pi$
$443$	16116.0i	1.72843i	−0.503123	+	0.864215i	$0.667816\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 774.000i	− 0.0818992i
$448$	0	0
$449$	−9018.00	−0.947852	−0.473926	−	0.880565i	$-0.657164\pi$
$449$	−9018.00	−0.947852	−0.473926	+	0.880565i	$0.657164\pi$
$450$	0	0
$451$	7920.00	0.826914
$452$	0	0
$453$	2424.00i	0.251412i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3670.00i	0.375657i	0.982202	+	0.187829i	$0.0601450\pi$
$457$	3670.00i	0.375657i	−0.982202	+	0.187829i	$0.939855\pi$
$458$	0	0
$459$	−1458.00	−0.148265
$460$	0	0
$461$	17562.0	1.77428	0.887141	−	0.461499i	$-0.152688\pi$
$461$	17562.0	1.77428	0.887141	+	0.461499i	$0.152688\pi$
$462$	0	0
$463$	− 1172.00i	− 0.117640i	−0.998269	−	0.0588202i	$-0.981266\pi$
$463$	− 1172.00i	− 0.117640i	0.998269	−	0.0588202i	$-0.0187338\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6876.00i	0.681335i	0.940184	+	0.340667i	$0.110653\pi$
$467$	6876.00i	0.681335i	−0.940184	+	0.340667i	$0.889347\pi$
$468$	0	0
$469$	3920.00	0.385946
$470$	0	0
$471$	7134.00	0.697914
$472$	0	0
$473$	− 2208.00i	− 0.214638i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4050.00i	− 0.388756i
$478$	0	0
$479$	2280.00	0.217486	0.108743	−	0.994070i	$-0.465317\pi$
$479$	2280.00	0.217486	0.108743	+	0.994070i	$0.465317\pi$
$480$	0	0
$481$	−5180.00	−0.491035
$482$	0	0
$483$	− 7200.00i	− 0.678284i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3076.00i	− 0.286215i	−0.989707	−	0.143108i	$-0.954290\pi$
$487$	− 3076.00i	− 0.286215i	0.989707	−	0.143108i	$-0.0457095\pi$
$488$	0	0
$489$	−156.000	−0.0144265
$490$	0	0
$491$	18912.0	1.73826	0.869131	−	0.494582i	$-0.164679\pi$
$491$	18912.0	1.73826	0.869131	+	0.494582i	$0.164679\pi$
$492$	0	0
$493$	− 4212.00i	− 0.384785i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5760.00i	0.519862i
$498$	0	0
$499$	9956.00	0.893170	0.446585	−	0.894741i	$-0.352640\pi$
$499$	9956.00	0.893170	0.446585	+	0.894741i	$0.352640\pi$
$500$	0	0
$501$	11160.0	0.995194
$502$	0	0
$503$	10656.0i	0.944588i	0.881441	+	0.472294i	$0.156574\pi$
$503$	10656.0i	0.944588i	−0.881441	+	0.472294i	$0.843426\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9837.00i	− 0.861689i
$508$	0	0
$509$	2766.00	0.240866	0.120433	−	0.992721i	$-0.461572\pi$
$509$	2766.00	0.240866	0.120433	+	0.992721i	$0.461572\pi$
$510$	0	0
$511$	8600.00	0.744504
$512$	0	0
$513$	3348.00i	0.288144i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 576.000i	− 0.0489989i
$518$	0	0
$519$	−1278.00	−0.108089
$520$	0	0
$521$	10530.0	0.885466	0.442733	−	0.896654i	$-0.354009\pi$
$521$	10530.0	0.885466	0.442733	+	0.896654i	$0.354009\pi$
$522$	0	0
$523$	− 12692.0i	− 1.06115i	−0.847637	−	0.530576i	$-0.821976\pi$
$523$	− 12692.0i	− 1.06115i	0.847637	−	0.530576i	$-0.178024\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10800.0i	0.892705i
$528$	0	0
$529$	−2233.00	−0.183529
$530$	0	0
$531$	−216.000	−0.0176527
$532$	0	0
$533$	24420.0i	1.98452i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4320.00i	− 0.347154i
$538$	0	0
$539$	−1368.00	−0.109321
$540$	0	0
$541$	18110.0	1.43920	0.719602	−	0.694386i	$-0.244324\pi$
$541$	18110.0	1.43920	0.719602	+	0.694386i	$0.244324\pi$
$542$	0	0
$543$	− 9390.00i	− 0.742106i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3620.00i	0.282962i	0.989941	+	0.141481i	$0.0451864\pi$
$547$	3620.00i	0.282962i	−0.989941	+	0.141481i	$0.954814\pi$
$548$	0	0
$549$	2898.00	0.225289
$550$	0	0
$551$	−9672.00	−0.747806
$552$	0	0
$553$	− 10400.0i	− 0.799734i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14166.0i	1.07762i	0.842428	+	0.538809i	$0.181125\pi$
$557$	14166.0i	1.07762i	−0.842428	+	0.538809i	$0.818875\pi$
$558$	0	0
$559$	6808.00	0.515112
$560$	0	0
$561$	3888.00	0.292605
$562$	0	0
$563$	13404.0i	1.00339i	0.865043	+	0.501697i	$0.167291\pi$
$563$	13404.0i	1.00339i	−0.865043	+	0.501697i	$0.832709\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1620.00i	0.119989i
$568$	0	0
$569$	18654.0	1.37437	0.687185	−	0.726483i	$-0.258846\pi$
$569$	18654.0	1.37437	0.687185	+	0.726483i	$0.258846\pi$
$570$	0	0
$571$	7684.00	0.563162	0.281581	−	0.959537i	$-0.409141\pi$
$571$	7684.00	0.563162	0.281581	+	0.959537i	$0.409141\pi$
$572$	0	0
$573$	− 10728.0i	− 0.782144i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1726.00i	0.124531i	0.998060	+	0.0622654i	$0.0198325\pi$
$577$	1726.00i	0.124531i	−0.998060	+	0.0622654i	$0.980167\pi$
$578$	0	0
$579$	−7998.00	−0.574068
$580$	0	0
$581$	3120.00	0.222787
$582$	0	0
$583$	10800.0i	0.767222i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10596.0i	0.745049i	0.928022	+	0.372524i	$0.121508\pi$
$587$	10596.0i	0.745049i	−0.928022	+	0.372524i	$0.878492\pi$
$588$	0	0
$589$	24800.0	1.73492
$590$	0	0
$591$	−8154.00	−0.567531
$592$	0	0
$593$	2862.00i	0.198193i	0.995078	+	0.0990963i	$0.0315952\pi$
$593$	2862.00i	0.198193i	−0.995078	+	0.0990963i	$0.968405\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11496.0i	− 0.788107i
$598$	0	0
$599$	−23592.0	−1.60925	−0.804627	−	0.593781i	$-0.797635\pi$
$599$	−23592.0	−1.60925	−0.804627	+	0.593781i	$0.797635\pi$
$600$	0	0
$601$	−9574.00	−0.649803	−0.324902	−	0.945748i	$-0.605331\pi$
$601$	−9574.00	−0.649803	−0.324902	+	0.945748i	$0.605331\pi$
$602$	0	0
$603$	1764.00i	0.119130i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17444.0i	1.16644i	0.812314	+	0.583221i	$0.198208\pi$
$607$	17444.0i	1.16644i	−0.812314	+	0.583221i	$0.801792\pi$
$608$	0	0
$609$	−4680.00	−0.311401
$610$	0	0
$611$	1776.00	0.117593
$612$	0	0
$613$	− 2374.00i	− 0.156419i	−0.996937	−	0.0782096i	$-0.975080\pi$
$613$	− 2374.00i	− 0.156419i	0.996937	−	0.0782096i	$-0.0249203\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12162.0i	0.793555i	0.917915	+	0.396778i	$0.129872\pi$
$617$	12162.0i	0.793555i	−0.917915	+	0.396778i	$0.870128\pi$
$618$	0	0
$619$	8804.00	0.571668	0.285834	−	0.958279i	$-0.407729\pi$
$619$	8804.00	0.571668	0.285834	+	0.958279i	$0.407729\pi$
$620$	0	0
$621$	3240.00	0.209367
$622$	0	0
$623$	− 20520.0i	− 1.31961i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8928.00i	− 0.568660i
$628$	0	0
$629$	3780.00	0.239616
$630$	0	0
$631$	12688.0	0.800478	0.400239	−	0.916411i	$-0.368927\pi$
$631$	12688.0	0.800478	0.400239	+	0.916411i	$0.368927\pi$
$632$	0	0
$633$	− 3300.00i	− 0.207209i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4218.00i	− 0.262360i
$638$	0	0
$639$	−2592.00	−0.160466
$640$	0	0
$641$	−9150.00	−0.563812	−0.281906	−	0.959442i	$-0.590967\pi$
$641$	−9150.00	−0.563812	−0.281906	+	0.959442i	$0.590967\pi$
$642$	0	0
$643$	− 25292.0i	− 1.55120i	−0.631227	−	0.775598i	$-0.717448\pi$
$643$	− 25292.0i	− 1.55120i	0.631227	−	0.775598i	$-0.282552\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2736.00i	− 0.166249i	−0.996539	−	0.0831246i	$-0.973510\pi$
$647$	− 2736.00i	− 0.166249i	0.996539	−	0.0831246i	$-0.0264900\pi$
$648$	0	0
$649$	576.000	0.0348382
$650$	0	0
$651$	12000.0	0.722453
$652$	0	0
$653$	22218.0i	1.33148i	0.746183	+	0.665741i	$0.231884\pi$
$653$	22218.0i	1.33148i	−0.746183	+	0.665741i	$0.768116\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3870.00i	0.229807i
$658$	0	0
$659$	14520.0	0.858299	0.429149	−	0.903234i	$-0.358813\pi$
$659$	14520.0	0.858299	0.429149	+	0.903234i	$0.358813\pi$
$660$	0	0
$661$	−10618.0	−0.624799	−0.312400	−	0.949951i	$-0.601133\pi$
$661$	−10618.0	−0.624799	−0.312400	+	0.949951i	$0.601133\pi$
$662$	0	0
$663$	11988.0i	0.702225i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9360.00i	0.543359i
$668$	0	0
$669$	5892.00	0.340505
$670$	0	0
$671$	−7728.00	−0.444614
$672$	0	0
$673$	1370.00i	0.0784690i	0.999230	+	0.0392345i	$0.0124919\pi$
$673$	1370.00i	0.0784690i	−0.999230	+	0.0392345i	$0.987508\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13758.0i	0.781038i	0.920595	+	0.390519i	$0.127704\pi$
$677$	13758.0i	0.781038i	−0.920595	+	0.390519i	$0.872296\pi$
$678$	0	0
$679$	−5720.00	−0.323289
$680$	0	0
$681$	−1980.00	−0.111415
$682$	0	0
$683$	− 11988.0i	− 0.671608i	−0.941932	−	0.335804i	$-0.890992\pi$
$683$	− 11988.0i	− 0.671608i	0.941932	−	0.335804i	$-0.109008\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5718.00i	0.317548i
$688$	0	0
$689$	−33300.0	−1.84126
$690$	0	0
$691$	−32996.0	−1.81654	−0.908268	−	0.418388i	$-0.862595\pi$
$691$	−32996.0	−1.81654	−0.908268	+	0.418388i	$0.862595\pi$
$692$	0	0
$693$	− 4320.00i	− 0.236801i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 17820.0i	− 0.968408i
$698$	0	0
$699$	4374.00	0.236681
$700$	0	0
$701$	−25902.0	−1.39558	−0.697792	−	0.716300i	$-0.745834\pi$
$701$	−25902.0	−1.39558	−0.697792	+	0.716300i	$0.745834\pi$
$702$	0	0
$703$	− 8680.00i	− 0.465679i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 34680.0i	− 1.84480i
$708$	0	0
$709$	27394.0	1.45106	0.725531	−	0.688189i	$-0.241594\pi$
$709$	27394.0	1.45106	0.725531	+	0.688189i	$0.241594\pi$
$710$	0	0
$711$	4680.00	0.246855
$712$	0	0
$713$	− 24000.0i	− 1.26060i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3528.00i	0.183760i
$718$	0	0
$719$	34848.0	1.80753	0.903763	−	0.428033i	$-0.140793\pi$
$719$	34848.0	1.80753	0.903763	+	0.428033i	$0.140793\pi$
$720$	0	0
$721$	9040.00	0.466945
$722$	0	0
$723$	2598.00i	0.133639i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28028.0i	1.42985i	0.699201	+	0.714925i	$0.253539\pi$
$727$	28028.0i	1.42985i	−0.699201	+	0.714925i	$0.746461\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−4968.00	−0.251365
$732$	0	0
$733$	18002.0i	0.907120i	0.891226	+	0.453560i	$0.149846\pi$
$733$	18002.0i	0.907120i	−0.891226	+	0.453560i	$0.850154\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4704.00i	− 0.235107i
$738$	0	0
$739$	15284.0	0.760800	0.380400	−	0.924822i	$-0.375786\pi$
$739$	15284.0	0.760800	0.380400	+	0.924822i	$0.375786\pi$
$740$	0	0
$741$	27528.0	1.36473
$742$	0	0
$743$	18768.0i	0.926691i	0.886178	+	0.463345i	$0.153351\pi$
$743$	18768.0i	0.926691i	−0.886178	+	0.463345i	$0.846649\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1404.00i	0.0687680i
$748$	0	0
$749$	28080.0	1.36985
$750$	0	0
$751$	−8696.00	−0.422532	−0.211266	−	0.977429i	$-0.567759\pi$
$751$	−8696.00	−0.422532	−0.211266	+	0.977429i	$0.567759\pi$
$752$	0	0
$753$	− 1296.00i	− 0.0627209i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38662.0i	1.85627i	0.372247	+	0.928134i	$0.378587\pi$
$757$	38662.0i	1.85627i	−0.372247	+	0.928134i	$0.621413\pi$
$758$	0	0
$759$	−8640.00	−0.413191
$760$	0	0
$761$	23874.0	1.13723	0.568615	−	0.822604i	$-0.307479\pi$
$761$	23874.0	1.13723	0.568615	+	0.822604i	$0.307479\pi$
$762$	0	0
$763$	29480.0i	1.39875i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1776.00i	0.0836084i
$768$	0	0
$769$	−23618.0	−1.10753	−0.553763	−	0.832675i	$-0.686808\pi$
$769$	−23618.0	−1.10753	−0.553763	+	0.832675i	$0.686808\pi$
$770$	0	0
$771$	7578.00	0.353975
$772$	0	0
$773$	11538.0i	0.536860i	0.963299	+	0.268430i	$0.0865049\pi$
$773$	11538.0i	0.536860i	−0.963299	+	0.268430i	$0.913495\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4200.00i	− 0.193918i
$778$	0	0
$779$	−40920.0	−1.88204
$780$	0	0
$781$	6912.00	0.316685
$782$	0	0
$783$	− 2106.00i	− 0.0961204i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 14884.0i	− 0.674152i	−0.941478	−	0.337076i	$-0.890562\pi$
$787$	− 14884.0i	− 0.674152i	0.941478	−	0.337076i	$-0.109438\pi$
$788$	0	0
$789$	16344.0	0.737467
$790$	0	0
$791$	−21720.0	−0.976327
$792$	0	0
$793$	− 23828.0i	− 1.06703i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11334.0i	0.503728i	0.967763	+	0.251864i	$0.0810435\pi$
$797$	11334.0i	0.503728i	−0.967763	+	0.251864i	$0.918957\pi$
$798$	0	0
$799$	−1296.00	−0.0573832
$800$	0	0
$801$	9234.00	0.407325
$802$	0	0
$803$	− 10320.0i	− 0.453530i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7722.00i	0.336837i
$808$	0	0
$809$	−44730.0	−1.94391	−0.971955	−	0.235167i	$-0.924436\pi$
$809$	−44730.0	−1.94391	−0.971955	+	0.235167i	$0.924436\pi$
$810$	0	0
$811$	42748.0	1.85091	0.925453	−	0.378862i	$-0.123684\pi$
$811$	42748.0	1.85091	0.925453	+	0.378862i	$0.123684\pi$
$812$	0	0
$813$	9552.00i	0.412058i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11408.0i	0.488513i
$818$	0	0
$819$	13320.0	0.568301
$820$	0	0
$821$	−31686.0	−1.34695	−0.673477	−	0.739208i	$-0.735200\pi$
$821$	−31686.0	−1.34695	−0.673477	+	0.739208i	$0.735200\pi$
$822$	0	0
$823$	− 11036.0i	− 0.467425i	−0.972306	−	0.233713i	$-0.924913\pi$
$823$	− 11036.0i	− 0.467425i	0.972306	−	0.233713i	$-0.0750875\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25884.0i	1.08836i	0.838968	+	0.544181i	$0.183159\pi$
$827$	25884.0i	1.08836i	−0.838968	+	0.544181i	$0.816841\pi$
$828$	0	0
$829$	−15950.0	−0.668234	−0.334117	−	0.942532i	$-0.608438\pi$
$829$	−15950.0	−0.668234	−0.334117	+	0.942532i	$0.608438\pi$
$830$	0	0
$831$	11886.0	0.496174
$832$	0	0
$833$	3078.00i	0.128027i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5400.00i	0.223000i
$838$	0	0
$839$	13800.0	0.567853	0.283927	−	0.958846i	$-0.408363\pi$
$839$	13800.0	0.567853	0.283927	+	0.958846i	$0.408363\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	0	0
$843$	− 24858.0i	− 1.01560i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 15100.0i	− 0.612565i
$848$	0	0
$849$	−8148.00	−0.329374
$850$	0	0
$851$	−8400.00	−0.338365
$852$	0	0
$853$	− 27862.0i	− 1.11838i	−0.829040	−	0.559189i	$-0.811113\pi$
$853$	− 27862.0i	− 1.11838i	0.829040	−	0.559189i	$-0.188887\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7314.00i	0.291530i	0.989319	+	0.145765i	$0.0465644\pi$
$857$	7314.00i	0.291530i	−0.989319	+	0.145765i	$0.953436\pi$
$858$	0	0
$859$	−28780.0	−1.14314	−0.571572	−	0.820552i	$-0.693666\pi$
$859$	−28780.0	−1.14314	−0.571572	+	0.820552i	$0.693666\pi$
$860$	0	0
$861$	−19800.0	−0.783719
$862$	0	0
$863$	32688.0i	1.28935i	0.764455	+	0.644677i	$0.223008\pi$
$863$	32688.0i	1.28935i	−0.764455	+	0.644677i	$0.776992\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5991.00i	0.234677i
$868$	0	0
$869$	−12480.0	−0.487175
$870$	0	0
$871$	14504.0	0.564236
$872$	0	0
$873$	− 2574.00i	− 0.0997900i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 36650.0i	− 1.41115i	−0.708633	−	0.705577i	$-0.750688\pi$
$877$	− 36650.0i	− 1.41115i	0.708633	−	0.705577i	$-0.249312\pi$
$878$	0	0
$879$	−18054.0	−0.692772
$880$	0	0
$881$	−2646.00	−0.101187	−0.0505936	−	0.998719i	$-0.516111\pi$
$881$	−2646.00	−0.101187	−0.0505936	+	0.998719i	$0.516111\pi$
$882$	0	0
$883$	− 10892.0i	− 0.415113i	−0.978223	−	0.207557i	$-0.933449\pi$
$883$	− 10892.0i	− 0.415113i	0.978223	−	0.207557i	$-0.0665511\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 43464.0i	− 1.64530i	−0.568550	−	0.822648i	$-0.692496\pi$
$887$	− 43464.0i	− 1.64530i	0.568550	−	0.822648i	$-0.307504\pi$
$888$	0	0
$889$	−24880.0	−0.938637
$890$	0	0
$891$	1944.00	0.0730937
$892$	0	0
$893$	2976.00i	0.111521i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 26640.0i	− 0.991621i
$898$	0	0
$899$	−15600.0	−0.578742
$900$	0	0
$901$	24300.0	0.898502
$902$	0	0
$903$	5520.00i	0.203426i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 14884.0i	− 0.544890i	−0.962171	−	0.272445i	$-0.912168\pi$
$907$	− 14884.0i	− 0.544890i	0.962171	−	0.272445i	$-0.0878323\pi$
$908$	0	0
$909$	15606.0	0.569437
$910$	0	0
$911$	1248.00	0.0453876	0.0226938	−	0.999742i	$-0.492776\pi$
$911$	1248.00	0.0453876	0.0226938	+	0.999742i	$0.492776\pi$
$912$	0	0
$913$	− 3744.00i	− 0.135716i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 46560.0i	− 1.67671i
$918$	0	0
$919$	−6640.00	−0.238339	−0.119169	−	0.992874i	$-0.538023\pi$
$919$	−6640.00	−0.238339	−0.119169	+	0.992874i	$0.538023\pi$
$920$	0	0
$921$	−27708.0	−0.991324
$922$	0	0
$923$	21312.0i	0.760014i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4068.00i	0.144132i
$928$	0	0
$929$	−29946.0	−1.05758	−0.528792	−	0.848751i	$-0.677355\pi$
$929$	−29946.0	−1.05758	−0.528792	+	0.848751i	$0.677355\pi$
$930$	0	0
$931$	7068.00	0.248812
$932$	0	0
$933$	− 4608.00i	− 0.161693i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 45002.0i	− 1.56900i	−0.620130	−	0.784499i	$-0.712920\pi$
$937$	− 45002.0i	− 1.56900i	0.620130	−	0.784499i	$-0.287080\pi$
$938$	0	0
$939$	22026.0	0.765486
$940$	0	0
$941$	6090.00	0.210976	0.105488	−	0.994421i	$-0.466360\pi$
$941$	6090.00	0.210976	0.105488	+	0.994421i	$0.466360\pi$
$942$	0	0
$943$	39600.0i	1.36750i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56388.0i	1.93491i	0.253035	+	0.967457i	$0.418571\pi$
$947$	56388.0i	1.93491i	−0.253035	+	0.967457i	$0.581429\pi$
$948$	0	0
$949$	31820.0	1.08843
$950$	0	0
$951$	−11682.0	−0.398333
$952$	0	0
$953$	10854.0i	0.368936i	0.982839	+	0.184468i	$0.0590561\pi$
$953$	10854.0i	0.368936i	−0.982839	+	0.184468i	$0.940944\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5616.00i	0.189696i
$958$	0	0
$959$	42360.0	1.42636
$960$	0	0
$961$	10209.0	0.342687
$962$	0	0
$963$	12636.0i	0.422834i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 42316.0i	− 1.40723i	−0.710582	−	0.703615i	$-0.751568\pi$
$967$	− 42316.0i	− 1.40723i	0.710582	−	0.703615i	$-0.248432\pi$
$968$	0	0
$969$	−20088.0	−0.665964
$970$	0	0
$971$	−24480.0	−0.809063	−0.404532	−	0.914524i	$-0.632565\pi$
$971$	−24480.0	−0.809063	−0.404532	+	0.914524i	$0.632565\pi$
$972$	0	0
$973$	46480.0i	1.53143i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6906.00i	0.226144i	0.993587	+	0.113072i	$0.0360690\pi$
$977$	6906.00i	0.226144i	−0.993587	+	0.113072i	$0.963931\pi$
$978$	0	0
$979$	−24624.0	−0.803868
$980$	0	0
$981$	−13266.0	−0.431754
$982$	0	0
$983$	− 6960.00i	− 0.225829i	−0.993605	−	0.112914i	$-0.963981\pi$
$983$	− 6960.00i	− 0.225829i	0.993605	−	0.112914i	$-0.0360186\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1440.00i	0.0464394i
$988$	0	0
$989$	11040.0	0.354956
$990$	0	0
$991$	−47792.0	−1.53195	−0.765975	−	0.642870i	$-0.777744\pi$
$991$	−47792.0	−1.53195	−0.765975	+	0.642870i	$0.777744\pi$
$992$	0	0
$993$	− 11076.0i	− 0.353964i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9938.00i	− 0.315687i	−0.987464	−	0.157843i	$-0.949546\pi$
$997$	− 9938.00i	− 0.315687i	0.987464	−	0.157843i	$-0.0504541\pi$
$998$	0	0
$999$	1890.00	0.0598568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.m.49.2		2
4.3	odd	2		75.4.b.a.49.1		2
5.2	odd	4		240.4.a.f.1.1		1
5.3	odd	4		1200.4.a.o.1.1		1
5.4	even	2	inner	1200.4.f.m.49.1		2
12.11	even	2		225.4.b.d.199.2		2
15.2	even	4		720.4.a.r.1.1		1
20.3	even	4		75.4.a.a.1.1		1
20.7	even	4		15.4.a.b.1.1	✓	1
20.19	odd	2		75.4.b.a.49.2		2
40.27	even	4		960.4.a.bi.1.1		1
40.37	odd	4		960.4.a.l.1.1		1
60.23	odd	4		225.4.a.g.1.1		1
60.47	odd	4		45.4.a.b.1.1		1
60.59	even	2		225.4.b.d.199.1		2
140.27	odd	4		735.4.a.i.1.1		1
180.7	even	12		405.4.e.d.271.1		2
180.47	odd	12		405.4.e.k.271.1		2
180.67	even	12		405.4.e.d.136.1		2
180.167	odd	12		405.4.e.k.136.1		2
220.87	odd	4		1815.4.a.a.1.1		1
420.167	even	4		2205.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	20.7	even	4
45.4.a.b.1.1		1	60.47	odd	4
75.4.a.a.1.1		1	20.3	even	4
75.4.b.a.49.1		2	4.3	odd	2
75.4.b.a.49.2		2	20.19	odd	2
225.4.a.g.1.1		1	60.23	odd	4
225.4.b.d.199.1		2	60.59	even	2
225.4.b.d.199.2		2	12.11	even	2
240.4.a.f.1.1		1	5.2	odd	4
405.4.e.d.136.1		2	180.67	even	12
405.4.e.d.271.1		2	180.7	even	12
405.4.e.k.136.1		2	180.167	odd	12
405.4.e.k.271.1		2	180.47	odd	12
720.4.a.r.1.1		1	15.2	even	4
735.4.a.i.1.1		1	140.27	odd	4
960.4.a.l.1.1		1	40.37	odd	4
960.4.a.bi.1.1		1	40.27	even	4
1200.4.a.o.1.1		1	5.3	odd	4
1200.4.f.m.49.1		2	5.4	even	2	inner
1200.4.f.m.49.2		2	1.1	even	1	trivial
1815.4.a.a.1.1		1	220.87	odd	4
2205.4.a.c.1.1		1	420.167	even	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +20.0000i q^{7} -9.00000 q^{9} +24.0000 q^{11} +74.0000i q^{13} -54.0000i q^{17} -124.000 q^{19} -60.0000 q^{21} +120.000i q^{23} -27.0000i q^{27} +78.0000 q^{29} -200.000 q^{31} +72.0000i q^{33} +70.0000i q^{37} -222.000 q^{39} +330.000 q^{41} -92.0000i q^{43} -24.0000i q^{47} -57.0000 q^{49} +162.000 q^{51} +450.000i q^{53} -372.000i q^{57} +24.0000 q^{59} -322.000 q^{61} -180.000i q^{63} -196.000i q^{67} -360.000 q^{69} +288.000 q^{71} -430.000i q^{73} +480.000i q^{77} -520.000 q^{79} +81.0000 q^{81} -156.000i q^{83} +234.000i q^{87} -1026.00 q^{89} -1480.00 q^{91} -600.000i q^{93} +286.000i q^{97} -216.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 48 q^{11} - 248 q^{19} - 120 q^{21} + 156 q^{29} - 400 q^{31} - 444 q^{39} + 660 q^{41} - 114 q^{49} + 324 q^{51} + 48 q^{59} - 644 q^{61} - 720 q^{69} + 576 q^{71} - 1040 q^{79} + 162 q^{81}+ \cdots - 432 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 48 * q^11 - 248 * q^19 - 120 * q^21 + 156 * q^29 - 400 * q^31 - 444 * q^39 + 660 * q^41 - 114 * q^49 + 324 * q^51 + 48 * q^59 - 644 * q^61 - 720 * q^69 + 576 * q^71 - 1040 * q^79 + 162 * q^81 - 2052 * q^89 - 2960 * q^91 - 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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