LMFDB - Embedded newform 2850.2.a.bm.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2850,2,Mod(1,2850)] 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("2850.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,-3,3,0,-3,0,3,3,0,8,-3,4,0,0,3,2,3,3,0,0,8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.7573645761$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2850.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.35026 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.962389 q^{11} -1.00000 q^{12} +1.61213 q^{13} +3.35026 q^{14} +1.00000 q^{16} +0.387873 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.35026 q^{21} -0.962389 q^{22} +0.962389 q^{23} -1.00000 q^{24} +1.61213 q^{26} -1.00000 q^{27} +3.35026 q^{28} +6.96239 q^{29} +3.35026 q^{31} +1.00000 q^{32} +0.962389 q^{33} +0.387873 q^{34} +1.00000 q^{36} +1.61213 q^{37} +1.00000 q^{38} -1.61213 q^{39} -9.27504 q^{41} -3.35026 q^{42} -6.18664 q^{43} -0.962389 q^{44} +0.962389 q^{46} -0.962389 q^{47} -1.00000 q^{48} +4.22425 q^{49} -0.387873 q^{51} +1.61213 q^{52} -6.00000 q^{53} -1.00000 q^{54} +3.35026 q^{56} -1.00000 q^{57} +6.96239 q^{58} +10.3127 q^{59} +11.9248 q^{61} +3.35026 q^{62} +3.35026 q^{63} +1.00000 q^{64} +0.962389 q^{66} +7.22425 q^{67} +0.387873 q^{68} -0.962389 q^{69} +7.22425 q^{71} +1.00000 q^{72} +3.22425 q^{73} +1.61213 q^{74} +1.00000 q^{76} -3.22425 q^{77} -1.61213 q^{78} -3.35026 q^{79} +1.00000 q^{81} -9.27504 q^{82} -15.0132 q^{83} -3.35026 q^{84} -6.18664 q^{86} -6.96239 q^{87} -0.962389 q^{88} +4.64974 q^{89} +5.40105 q^{91} +0.962389 q^{92} -3.35026 q^{93} -0.962389 q^{94} -1.00000 q^{96} +10.9624 q^{97} +4.22425 q^{98} -0.962389 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 8 q^{11} - 3 q^{12} + 4 q^{13} + 3 q^{16} + 2 q^{17} + 3 q^{18} + 3 q^{19} + 8 q^{22} - 8 q^{23} - 3 q^{24} + 4 q^{26} - 3 q^{27} + 10 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 8 * q^11 - 3 * q^12 + 4 * q^13 + 3 * q^16 + 2 * q^17 + 3 * q^18 + 3 * q^19 + 8 * q^22 - 8 * q^23 - 3 * q^24 + 4 * q^26 - 3 * q^27 + 10 * q^29 + 3 * q^32 - 8 * q^33 + 2 * q^34 + 3 * q^36 + 4 * q^37 + 3 * q^38 - 4 * q^39 + 4 * q^41 - 6 * q^43 + 8 * q^44 - 8 * q^46 + 8 * q^47 - 3 * q^48 + 11 * q^49 - 2 * q^51 + 4 * q^52 - 18 * q^53 - 3 * q^54 - 3 * q^57 + 10 * q^58 + 10 * q^59 + 14 * q^61 + 3 * q^64 - 8 * q^66 + 20 * q^67 + 2 * q^68 + 8 * q^69 + 20 * q^71 + 3 * q^72 + 8 * q^73 + 4 * q^74 + 3 * q^76 - 8 * q^77 - 4 * q^78 + 3 * q^81 + 4 * q^82 - 4 * q^83 - 6 * q^86 - 10 * q^87 + 8 * q^88 + 24 * q^89 - 24 * q^91 - 8 * q^92 + 8 * q^94 - 3 * q^96 + 22 * q^97 + 11 * q^98 + 8 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	3.35026	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$7$	3.35026	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.962389	−0.290171	−0.145086	−	0.989419i	$-0.546346\pi$
$11$	−0.962389	−0.290171	−0.145086	+	0.989419i	$0.546346\pi$
$12$	−1.00000	−0.288675
$13$	1.61213	0.447124	0.223562	−	0.974690i	$-0.428232\pi$
$13$	1.61213	0.447124	0.223562	+	0.974690i	$0.428232\pi$
$14$	3.35026	0.895395
$15$	0	0
$16$	1.00000	0.250000
$17$	0.387873	0.0940731	0.0470365	−	0.998893i	$-0.485022\pi$
$17$	0.387873	0.0940731	0.0470365	+	0.998893i	$0.485022\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.35026	−0.731087
$22$	−0.962389	−0.205182
$23$	0.962389	0.200672	0.100336	−	0.994954i	$-0.468008\pi$
$23$	0.962389	0.200672	0.100336	+	0.994954i	$0.468008\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.61213	0.316164
$27$	−1.00000	−0.192450
$28$	3.35026	0.633140
$29$	6.96239	1.29288	0.646442	−	0.762964i	$-0.276256\pi$
$29$	6.96239	1.29288	0.646442	+	0.762964i	$0.276256\pi$
$30$	0	0
$31$	3.35026	0.601725	0.300862	−	0.953668i	$-0.402726\pi$
$31$	3.35026	0.601725	0.300862	+	0.953668i	$0.402726\pi$
$32$	1.00000	0.176777
$33$	0.962389	0.167530
$34$	0.387873	0.0665197
$35$	0	0
$36$	1.00000	0.166667
$37$	1.61213	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$37$	1.61213	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$38$	1.00000	0.162221
$39$	−1.61213	−0.258147
$40$	0	0
$41$	−9.27504	−1.44852	−0.724259	−	0.689528i	$-0.757818\pi$
$41$	−9.27504	−1.44852	−0.724259	+	0.689528i	$0.757818\pi$
$42$	−3.35026	−0.516957
$43$	−6.18664	−0.943454	−0.471727	−	0.881745i	$-0.656369\pi$
$43$	−6.18664	−0.943454	−0.471727	+	0.881745i	$0.656369\pi$
$44$	−0.962389	−0.145086
$45$	0	0
$46$	0.962389	0.141896
$47$	−0.962389	−0.140379	−0.0701894	−	0.997534i	$-0.522360\pi$
$47$	−0.962389	−0.140379	−0.0701894	+	0.997534i	$0.522360\pi$
$48$	−1.00000	−0.144338
$49$	4.22425	0.603465
$50$	0	0
$51$	−0.387873	−0.0543131
$52$	1.61213	0.223562
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.35026	0.447698
$57$	−1.00000	−0.132453
$58$	6.96239	0.914206
$59$	10.3127	1.34259	0.671296	−	0.741189i	$-0.265738\pi$
$59$	10.3127	1.34259	0.671296	+	0.741189i	$0.265738\pi$
$60$	0	0
$61$	11.9248	1.52681	0.763406	−	0.645919i	$-0.223526\pi$
$61$	11.9248	1.52681	0.763406	+	0.645919i	$0.223526\pi$
$62$	3.35026	0.425484
$63$	3.35026	0.422093
$64$	1.00000	0.125000
$65$	0	0
$66$	0.962389	0.118462
$67$	7.22425	0.882583	0.441292	−	0.897364i	$-0.354520\pi$
$67$	7.22425	0.882583	0.441292	+	0.897364i	$0.354520\pi$
$68$	0.387873	0.0470365
$69$	−0.962389	−0.115858
$70$	0	0
$71$	7.22425	0.857361	0.428681	−	0.903456i	$-0.358979\pi$
$71$	7.22425	0.857361	0.428681	+	0.903456i	$0.358979\pi$
$72$	1.00000	0.117851
$73$	3.22425	0.377370	0.188685	−	0.982038i	$-0.439577\pi$
$73$	3.22425	0.377370	0.188685	+	0.982038i	$0.439577\pi$
$74$	1.61213	0.187406
$75$	0	0
$76$	1.00000	0.114708
$77$	−3.22425	−0.367438
$78$	−1.61213	−0.182537
$79$	−3.35026	−0.376934	−0.188467	−	0.982080i	$-0.560352\pi$
$79$	−3.35026	−0.376934	−0.188467	+	0.982080i	$0.560352\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.27504	−1.02426
$83$	−15.0132	−1.64791	−0.823955	−	0.566655i	$-0.808237\pi$
$83$	−15.0132	−1.64791	−0.823955	+	0.566655i	$0.808237\pi$
$84$	−3.35026	−0.365544
$85$	0	0
$86$	−6.18664	−0.667123
$87$	−6.96239	−0.746446
$88$	−0.962389	−0.102591
$89$	4.64974	0.492871	0.246436	−	0.969159i	$-0.420741\pi$
$89$	4.64974	0.492871	0.246436	+	0.969159i	$0.420741\pi$
$90$	0	0
$91$	5.40105	0.566184
$92$	0.962389	0.100336
$93$	−3.35026	−0.347406
$94$	−0.962389	−0.0992628
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.9624	1.11306	0.556531	−	0.830827i	$-0.312132\pi$
$97$	10.9624	1.11306	0.556531	+	0.830827i	$0.312132\pi$
$98$	4.22425	0.426714
$99$	−0.962389	−0.0967237
$100$	0	0
$101$	2.72496	0.271144	0.135572	−	0.990768i	$-0.456713\pi$
$101$	2.72496	0.271144	0.135572	+	0.990768i	$0.456713\pi$
$102$	−0.387873	−0.0384052
$103$	−0.574515	−0.0566087	−0.0283043	−	0.999599i	$-0.509011\pi$
$103$	−0.574515	−0.0566087	−0.0283043	+	0.999599i	$0.509011\pi$
$104$	1.61213	0.158082
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−10.7005	−1.03446	−0.517229	−	0.855847i	$-0.673037\pi$
$107$	−10.7005	−1.03446	−0.517229	+	0.855847i	$0.673037\pi$
$108$	−1.00000	−0.0962250
$109$	10.1260	0.969896	0.484948	−	0.874543i	$-0.338839\pi$
$109$	10.1260	0.969896	0.484948	+	0.874543i	$0.338839\pi$
$110$	0	0
$111$	−1.61213	−0.153016
$112$	3.35026	0.316570
$113$	−20.5501	−1.93319	−0.966594	−	0.256311i	$-0.917493\pi$
$113$	−20.5501	−1.93319	−0.966594	+	0.256311i	$0.917493\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	6.96239	0.646442
$117$	1.61213	0.149041
$118$	10.3127	0.949356
$119$	1.29948	0.119123
$120$	0	0
$121$	−10.0738	−0.915801
$122$	11.9248	1.07962
$123$	9.27504	0.836302
$124$	3.35026	0.300862
$125$	0	0
$126$	3.35026	0.298465
$127$	−1.35026	−0.119816	−0.0599082	−	0.998204i	$-0.519081\pi$
$127$	−1.35026	−0.119816	−0.0599082	+	0.998204i	$0.519081\pi$
$128$	1.00000	0.0883883
$129$	6.18664	0.544703
$130$	0	0
$131$	12.4387	1.08677	0.543385	−	0.839483i	$-0.317142\pi$
$131$	12.4387	1.08677	0.543385	+	0.839483i	$0.317142\pi$
$132$	0.962389	0.0837652
$133$	3.35026	0.290505
$134$	7.22425	0.624080
$135$	0	0
$136$	0.387873	0.0332598
$137$	18.1622	1.55170	0.775851	−	0.630916i	$-0.217321\pi$
$137$	18.1622	1.55170	0.775851	+	0.630916i	$0.217321\pi$
$138$	−0.962389	−0.0819240
$139$	8.77575	0.744349	0.372175	−	0.928163i	$-0.378612\pi$
$139$	8.77575	0.744349	0.372175	+	0.928163i	$0.378612\pi$
$140$	0	0
$141$	0.962389	0.0810477
$142$	7.22425	0.606246
$143$	−1.55149	−0.129742
$144$	1.00000	0.0833333
$145$	0	0
$146$	3.22425	0.266841
$147$	−4.22425	−0.348411
$148$	1.61213	0.132516
$149$	15.9756	1.30877	0.654385	−	0.756162i	$-0.272928\pi$
$149$	15.9756	1.30877	0.654385	+	0.756162i	$0.272928\pi$
$150$	0	0
$151$	18.4241	1.49933	0.749665	−	0.661818i	$-0.230215\pi$
$151$	18.4241	1.49933	0.749665	+	0.661818i	$0.230215\pi$
$152$	1.00000	0.0811107
$153$	0.387873	0.0313577
$154$	−3.22425	−0.259818
$155$	0	0
$156$	−1.61213	−0.129073
$157$	13.7889	1.10048	0.550238	−	0.835008i	$-0.314537\pi$
$157$	13.7889	1.10048	0.550238	+	0.835008i	$0.314537\pi$
$158$	−3.35026	−0.266533
$159$	6.00000	0.475831
$160$	0	0
$161$	3.22425	0.254107
$162$	1.00000	0.0785674
$163$	3.73813	0.292793	0.146397	−	0.989226i	$-0.453232\pi$
$163$	3.73813	0.292793	0.146397	+	0.989226i	$0.453232\pi$
$164$	−9.27504	−0.724259
$165$	0	0
$166$	−15.0132	−1.16525
$167$	−15.4763	−1.19759	−0.598795	−	0.800902i	$-0.704354\pi$
$167$	−15.4763	−1.19759	−0.598795	+	0.800902i	$0.704354\pi$
$168$	−3.35026	−0.258478
$169$	−10.4010	−0.800081
$170$	0	0
$171$	1.00000	0.0764719
$172$	−6.18664	−0.471727
$173$	1.47627	0.112239	0.0561194	−	0.998424i	$-0.482127\pi$
$173$	1.47627	0.112239	0.0561194	+	0.998424i	$0.482127\pi$
$174$	−6.96239	−0.527817
$175$	0	0
$176$	−0.962389	−0.0725428
$177$	−10.3127	−0.775146
$178$	4.64974	0.348513
$179$	−14.3127	−1.06978	−0.534889	−	0.844922i	$-0.679647\pi$
$179$	−14.3127	−1.06978	−0.534889	+	0.844922i	$0.679647\pi$
$180$	0	0
$181$	−8.82653	−0.656071	−0.328035	−	0.944665i	$-0.606387\pi$
$181$	−8.82653	−0.656071	−0.328035	+	0.944665i	$0.606387\pi$
$182$	5.40105	0.400352
$183$	−11.9248	−0.881505
$184$	0.962389	0.0709482
$185$	0	0
$186$	−3.35026	−0.245653
$187$	−0.373285	−0.0272973
$188$	−0.962389	−0.0701894
$189$	−3.35026	−0.243696
$190$	0	0
$191$	2.31265	0.167338	0.0836688	−	0.996494i	$-0.473336\pi$
$191$	2.31265	0.167338	0.0836688	+	0.996494i	$0.473336\pi$
$192$	−1.00000	−0.0721688
$193$	−7.58769	−0.546174	−0.273087	−	0.961989i	$-0.588045\pi$
$193$	−7.58769	−0.546174	−0.273087	+	0.961989i	$0.588045\pi$
$194$	10.9624	0.787054
$195$	0	0
$196$	4.22425	0.301732
$197$	18.8119	1.34030	0.670148	−	0.742228i	$-0.266231\pi$
$197$	18.8119	1.34030	0.670148	+	0.742228i	$0.266231\pi$
$198$	−0.962389	−0.0683940
$199$	−9.40105	−0.666423	−0.333211	−	0.942852i	$-0.608132\pi$
$199$	−9.40105	−0.666423	−0.333211	+	0.942852i	$0.608132\pi$
$200$	0	0
$201$	−7.22425	−0.509560
$202$	2.72496	0.191728
$203$	23.3258	1.63715
$204$	−0.387873	−0.0271566
$205$	0	0
$206$	−0.574515	−0.0400284
$207$	0.962389	0.0668906
$208$	1.61213	0.111781
$209$	−0.962389	−0.0665698
$210$	0	0
$211$	4.77575	0.328776	0.164388	−	0.986396i	$-0.447435\pi$
$211$	4.77575	0.328776	0.164388	+	0.986396i	$0.447435\pi$
$212$	−6.00000	−0.412082
$213$	−7.22425	−0.494998
$214$	−10.7005	−0.731473
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	11.2243	0.761952
$218$	10.1260	0.685820
$219$	−3.22425	−0.217875
$220$	0	0
$221$	0.625301	0.0420623
$222$	−1.61213	−0.108199
$223$	24.6761	1.65243	0.826216	−	0.563353i	$-0.190489\pi$
$223$	24.6761	1.65243	0.826216	+	0.563353i	$0.190489\pi$
$224$	3.35026	0.223849
$225$	0	0
$226$	−20.5501	−1.36697
$227$	−15.4763	−1.02720	−0.513598	−	0.858031i	$-0.671688\pi$
$227$	−15.4763	−1.02720	−0.513598	+	0.858031i	$0.671688\pi$
$228$	−1.00000	−0.0662266
$229$	21.3258	1.40925	0.704625	−	0.709580i	$-0.251115\pi$
$229$	21.3258	1.40925	0.704625	+	0.709580i	$0.251115\pi$
$230$	0	0
$231$	3.22425	0.212140
$232$	6.96239	0.457103
$233$	9.01317	0.590473	0.295236	−	0.955424i	$-0.404602\pi$
$233$	9.01317	0.590473	0.295236	+	0.955424i	$0.404602\pi$
$234$	1.61213	0.105388
$235$	0	0
$236$	10.3127	0.671296
$237$	3.35026	0.217623
$238$	1.29948	0.0842326
$239$	−0.135857	−0.00878787	−0.00439393	−	0.999990i	$-0.501399\pi$
$239$	−0.135857	−0.00878787	−0.00439393	+	0.999990i	$0.501399\pi$
$240$	0	0
$241$	−25.8496	−1.66512	−0.832558	−	0.553938i	$-0.813125\pi$
$241$	−25.8496	−1.66512	−0.832558	+	0.553938i	$0.813125\pi$
$242$	−10.0738	−0.647569
$243$	−1.00000	−0.0641500
$244$	11.9248	0.763406
$245$	0	0
$246$	9.27504	0.591355
$247$	1.61213	0.102577
$248$	3.35026	0.212742
$249$	15.0132	0.951421
$250$	0	0
$251$	−10.1114	−0.638227	−0.319114	−	0.947716i	$-0.603385\pi$
$251$	−10.1114	−0.638227	−0.319114	+	0.947716i	$0.603385\pi$
$252$	3.35026	0.211047
$253$	−0.926192	−0.0582292
$254$	−1.35026	−0.0847230
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.9986	−1.68413	−0.842063	−	0.539380i	$-0.818659\pi$
$257$	−26.9986	−1.68413	−0.842063	+	0.539380i	$0.818659\pi$
$258$	6.18664	0.385164
$259$	5.40105	0.335605
$260$	0	0
$261$	6.96239	0.430961
$262$	12.4387	0.768463
$263$	−15.0376	−0.927259	−0.463629	−	0.886029i	$-0.653453\pi$
$263$	−15.0376	−0.927259	−0.463629	+	0.886029i	$0.653453\pi$
$264$	0.962389	0.0592309
$265$	0	0
$266$	3.35026	0.205418
$267$	−4.64974	−0.284559
$268$	7.22425	0.441292
$269$	−4.51388	−0.275216	−0.137608	−	0.990487i	$-0.543941\pi$
$269$	−4.51388	−0.275216	−0.137608	+	0.990487i	$0.543941\pi$
$270$	0	0
$271$	15.8496	0.962792	0.481396	−	0.876503i	$-0.340130\pi$
$271$	15.8496	0.962792	0.481396	+	0.876503i	$0.340130\pi$
$272$	0.387873	0.0235183
$273$	−5.40105	−0.326886
$274$	18.1622	1.09722
$275$	0	0
$276$	−0.962389	−0.0579290
$277$	−10.3127	−0.619627	−0.309814	−	0.950797i	$-0.600267\pi$
$277$	−10.3127	−0.619627	−0.309814	+	0.950797i	$0.600267\pi$
$278$	8.77575	0.526334
$279$	3.35026	0.200575
$280$	0	0
$281$	24.3488	1.45253	0.726265	−	0.687415i	$-0.241254\pi$
$281$	24.3488	1.45253	0.726265	+	0.687415i	$0.241254\pi$
$282$	0.962389	0.0573094
$283$	−26.2882	−1.56267	−0.781336	−	0.624111i	$-0.785461\pi$
$283$	−26.2882	−1.56267	−0.781336	+	0.624111i	$0.785461\pi$
$284$	7.22425	0.428681
$285$	0	0
$286$	−1.55149	−0.0917417
$287$	−31.0738	−1.83423
$288$	1.00000	0.0589256
$289$	−16.8496	−0.991150
$290$	0	0
$291$	−10.9624	−0.642627
$292$	3.22425	0.188685
$293$	13.0738	0.763780	0.381890	−	0.924208i	$-0.375273\pi$
$293$	13.0738	0.763780	0.381890	+	0.924208i	$0.375273\pi$
$294$	−4.22425	−0.246363
$295$	0	0
$296$	1.61213	0.0937030
$297$	0.962389	0.0558435
$298$	15.9756	0.925439
$299$	1.55149	0.0897251
$300$	0	0
$301$	−20.7269	−1.19468
$302$	18.4241	1.06019
$303$	−2.72496	−0.156545
$304$	1.00000	0.0573539
$305$	0	0
$306$	0.387873	0.0221732
$307$	7.07381	0.403724	0.201862	−	0.979414i	$-0.435301\pi$
$307$	7.07381	0.403724	0.201862	+	0.979414i	$0.435301\pi$
$308$	−3.22425	−0.183719
$309$	0.574515	0.0326830
$310$	0	0
$311$	26.3127	1.49205	0.746027	−	0.665916i	$-0.231959\pi$
$311$	26.3127	1.49205	0.746027	+	0.665916i	$0.231959\pi$
$312$	−1.61213	−0.0912687
$313$	18.7005	1.05702	0.528508	−	0.848928i	$-0.322752\pi$
$313$	18.7005	1.05702	0.528508	+	0.848928i	$0.322752\pi$
$314$	13.7889	0.778154
$315$	0	0
$316$	−3.35026	−0.188467
$317$	−26.4749	−1.48698	−0.743488	−	0.668749i	$-0.766830\pi$
$317$	−26.4749	−1.48698	−0.743488	+	0.668749i	$0.766830\pi$
$318$	6.00000	0.336463
$319$	−6.70052	−0.375157
$320$	0	0
$321$	10.7005	0.597245
$322$	3.22425	0.179681
$323$	0.387873	0.0215818
$324$	1.00000	0.0555556
$325$	0	0
$326$	3.73813	0.207036
$327$	−10.1260	−0.559970
$328$	−9.27504	−0.512128
$329$	−3.22425	−0.177759
$330$	0	0
$331$	−30.7005	−1.68745	−0.843727	−	0.536773i	$-0.819643\pi$
$331$	−30.7005	−1.68745	−0.843727	+	0.536773i	$0.819643\pi$
$332$	−15.0132	−0.823955
$333$	1.61213	0.0883440
$334$	−15.4763	−0.846824
$335$	0	0
$336$	−3.35026	−0.182772
$337$	−6.81194	−0.371070	−0.185535	−	0.982638i	$-0.559402\pi$
$337$	−6.81194	−0.371070	−0.185535	+	0.982638i	$0.559402\pi$
$338$	−10.4010	−0.565742
$339$	20.5501	1.11613
$340$	0	0
$341$	−3.22425	−0.174603
$342$	1.00000	0.0540738
$343$	−9.29948	−0.502125
$344$	−6.18664	−0.333561
$345$	0	0
$346$	1.47627	0.0793648
$347$	−18.3879	−0.987113	−0.493556	−	0.869714i	$-0.664303\pi$
$347$	−18.3879	−0.987113	−0.493556	+	0.869714i	$0.664303\pi$
$348$	−6.96239	−0.373223
$349$	31.1490	1.66737	0.833685	−	0.552241i	$-0.186227\pi$
$349$	31.1490	1.66737	0.833685	+	0.552241i	$0.186227\pi$
$350$	0	0
$351$	−1.61213	−0.0860490
$352$	−0.962389	−0.0512955
$353$	−7.61213	−0.405153	−0.202576	−	0.979266i	$-0.564931\pi$
$353$	−7.61213	−0.405153	−0.202576	+	0.979266i	$0.564931\pi$
$354$	−10.3127	−0.548111
$355$	0	0
$356$	4.64974	0.246436
$357$	−1.29948	−0.0687756
$358$	−14.3127	−0.756447
$359$	−35.3112	−1.86366	−0.931828	−	0.362901i	$-0.881786\pi$
$359$	−35.3112	−1.86366	−0.931828	+	0.362901i	$0.881786\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−8.82653	−0.463912
$363$	10.0738	0.528738
$364$	5.40105	0.283092
$365$	0	0
$366$	−11.9248	−0.623318
$367$	−31.9756	−1.66911	−0.834555	−	0.550924i	$-0.814275\pi$
$367$	−31.9756	−1.66911	−0.834555	+	0.550924i	$0.814275\pi$
$368$	0.962389	0.0501680
$369$	−9.27504	−0.482839
$370$	0	0
$371$	−20.1016	−1.04362
$372$	−3.35026	−0.173703
$373$	−26.4894	−1.37157	−0.685786	−	0.727804i	$-0.740541\pi$
$373$	−26.4894	−1.37157	−0.685786	+	0.727804i	$0.740541\pi$
$374$	−0.373285	−0.0193021
$375$	0	0
$376$	−0.962389	−0.0496314
$377$	11.2243	0.578078
$378$	−3.35026	−0.172319
$379$	−19.3258	−0.992701	−0.496350	−	0.868122i	$-0.665327\pi$
$379$	−19.3258	−0.992701	−0.496350	+	0.868122i	$0.665327\pi$
$380$	0	0
$381$	1.35026	0.0691760
$382$	2.31265	0.118325
$383$	−3.37470	−0.172439	−0.0862195	−	0.996276i	$-0.527479\pi$
$383$	−3.37470	−0.172439	−0.0862195	+	0.996276i	$0.527479\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−7.58769	−0.386203
$387$	−6.18664	−0.314485
$388$	10.9624	0.556531
$389$	−11.3503	−0.575481	−0.287741	−	0.957708i	$-0.592904\pi$
$389$	−11.3503	−0.575481	−0.287741	+	0.957708i	$0.592904\pi$
$390$	0	0
$391$	0.373285	0.0188778
$392$	4.22425	0.213357
$393$	−12.4387	−0.627447
$394$	18.8119	0.947732
$395$	0	0
$396$	−0.962389	−0.0483618
$397$	18.8364	0.945371	0.472685	−	0.881231i	$-0.343285\pi$
$397$	18.8364	0.945371	0.472685	+	0.881231i	$0.343285\pi$
$398$	−9.40105	−0.471232
$399$	−3.35026	−0.167723
$400$	0	0
$401$	−4.12601	−0.206043	−0.103022	−	0.994679i	$-0.532851\pi$
$401$	−4.12601	−0.206043	−0.103022	+	0.994679i	$0.532851\pi$
$402$	−7.22425	−0.360313
$403$	5.40105	0.269045
$404$	2.72496	0.135572
$405$	0	0
$406$	23.3258	1.15764
$407$	−1.55149	−0.0769046
$408$	−0.387873	−0.0192026
$409$	2.52373	0.124790	0.0623952	−	0.998052i	$-0.480126\pi$
$409$	2.52373	0.124790	0.0623952	+	0.998052i	$0.480126\pi$
$410$	0	0
$411$	−18.1622	−0.895875
$412$	−0.574515	−0.0283043
$413$	34.5501	1.70010
$414$	0.962389	0.0472988
$415$	0	0
$416$	1.61213	0.0790410
$417$	−8.77575	−0.429750
$418$	−0.962389	−0.0470720
$419$	7.51247	0.367008	0.183504	−	0.983019i	$-0.441256\pi$
$419$	7.51247	0.367008	0.183504	+	0.983019i	$0.441256\pi$
$420$	0	0
$421$	3.67750	0.179230	0.0896152	−	0.995976i	$-0.471436\pi$
$421$	3.67750	0.179230	0.0896152	+	0.995976i	$0.471436\pi$
$422$	4.77575	0.232480
$423$	−0.962389	−0.0467929
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−7.22425	−0.350016
$427$	39.9511	1.93337
$428$	−10.7005	−0.517229
$429$	1.55149	0.0749068
$430$	0	0
$431$	−10.3272	−0.497446	−0.248723	−	0.968575i	$-0.580011\pi$
$431$	−10.3272	−0.497446	−0.248723	+	0.968575i	$0.580011\pi$
$432$	−1.00000	−0.0481125
$433$	−31.5877	−1.51801	−0.759004	−	0.651086i	$-0.774314\pi$
$433$	−31.5877	−1.51801	−0.759004	+	0.651086i	$0.774314\pi$
$434$	11.2243	0.538781
$435$	0	0
$436$	10.1260	0.484948
$437$	0.962389	0.0460373
$438$	−3.22425	−0.154061
$439$	−38.1524	−1.82091	−0.910456	−	0.413605i	$-0.864269\pi$
$439$	−38.1524	−1.82091	−0.910456	+	0.413605i	$0.864269\pi$
$440$	0	0
$441$	4.22425	0.201155
$442$	0.625301	0.0297425
$443$	−16.3127	−0.775037	−0.387519	−	0.921862i	$-0.626668\pi$
$443$	−16.3127	−0.775037	−0.387519	+	0.921862i	$0.626668\pi$
$444$	−1.61213	−0.0765082
$445$	0	0
$446$	24.6761	1.16845
$447$	−15.9756	−0.755618
$448$	3.35026	0.158285
$449$	−37.5271	−1.77101	−0.885506	−	0.464629i	$-0.846188\pi$
$449$	−37.5271	−1.77101	−0.885506	+	0.464629i	$0.846188\pi$
$450$	0	0
$451$	8.92619	0.420318
$452$	−20.5501	−0.966594
$453$	−18.4241	−0.865638
$454$	−15.4763	−0.726337
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	21.3258	0.996490
$459$	−0.387873	−0.0181044
$460$	0	0
$461$	12.3780	0.576502	0.288251	−	0.957555i	$-0.406926\pi$
$461$	12.3780	0.576502	0.288251	+	0.957555i	$0.406926\pi$
$462$	3.22425	0.150006
$463$	−32.4504	−1.50810	−0.754049	−	0.656818i	$-0.771902\pi$
$463$	−32.4504	−1.50810	−0.754049	+	0.656818i	$0.771902\pi$
$464$	6.96239	0.323221
$465$	0	0
$466$	9.01317	0.417527
$467$	−7.53690	−0.348766	−0.174383	−	0.984678i	$-0.555793\pi$
$467$	−7.53690	−0.348766	−0.174383	+	0.984678i	$0.555793\pi$
$468$	1.61213	0.0745206
$469$	24.2031	1.11760
$470$	0	0
$471$	−13.7889	−0.635360
$472$	10.3127	0.474678
$473$	5.95395	0.273763
$474$	3.35026	0.153883
$475$	0	0
$476$	1.29948	0.0595614
$477$	−6.00000	−0.274721
$478$	−0.135857	−0.00621396
$479$	15.2097	0.694947	0.347474	−	0.937690i	$-0.387040\pi$
$479$	15.2097	0.694947	0.347474	+	0.937690i	$0.387040\pi$
$480$	0	0
$481$	2.59895	0.118502
$482$	−25.8496	−1.17741
$483$	−3.22425	−0.146709
$484$	−10.0738	−0.457900
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	1.19982	0.0543689	0.0271844	−	0.999630i	$-0.491346\pi$
$487$	1.19982	0.0543689	0.0271844	+	0.999630i	$0.491346\pi$
$488$	11.9248	0.539809
$489$	−3.73813	−0.169044
$490$	0	0
$491$	−5.11283	−0.230739	−0.115369	−	0.993323i	$-0.536805\pi$
$491$	−5.11283	−0.230739	−0.115369	+	0.993323i	$0.536805\pi$
$492$	9.27504	0.418151
$493$	2.70052	0.121625
$494$	1.61213	0.0725330
$495$	0	0
$496$	3.35026	0.150431
$497$	24.2031	1.08566
$498$	15.0132	0.672756
$499$	14.2981	0.640069	0.320035	−	0.947406i	$-0.396305\pi$
$499$	14.2981	0.640069	0.320035	+	0.947406i	$0.396305\pi$
$500$	0	0
$501$	15.4763	0.691429
$502$	−10.1114	−0.451295
$503$	−11.5125	−0.513316	−0.256658	−	0.966502i	$-0.582621\pi$
$503$	−11.5125	−0.513316	−0.256658	+	0.966502i	$0.582621\pi$
$504$	3.35026	0.149233
$505$	0	0
$506$	−0.926192	−0.0411742
$507$	10.4010	0.461927
$508$	−1.35026	−0.0599082
$509$	−31.9610	−1.41665	−0.708323	−	0.705889i	$-0.750548\pi$
$509$	−31.9610	−1.41665	−0.708323	+	0.705889i	$0.750548\pi$
$510$	0	0
$511$	10.8021	0.477857
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−26.9986	−1.19086
$515$	0	0
$516$	6.18664	0.272352
$517$	0.926192	0.0407339
$518$	5.40105	0.237308
$519$	−1.47627	−0.0648010
$520$	0	0
$521$	−33.2750	−1.45781	−0.728903	−	0.684617i	$-0.759969\pi$
$521$	−33.2750	−1.45781	−0.728903	+	0.684617i	$0.759969\pi$
$522$	6.96239	0.304735
$523$	9.29948	0.406638	0.203319	−	0.979113i	$-0.434827\pi$
$523$	9.29948	0.406638	0.203319	+	0.979113i	$0.434827\pi$
$524$	12.4387	0.543385
$525$	0	0
$526$	−15.0376	−0.655671
$527$	1.29948	0.0566061
$528$	0.962389	0.0418826
$529$	−22.0738	−0.959731
$530$	0	0
$531$	10.3127	0.447531
$532$	3.35026	0.145252
$533$	−14.9525	−0.647666
$534$	−4.64974	−0.201214
$535$	0	0
$536$	7.22425	0.312040
$537$	14.3127	0.617636
$538$	−4.51388	−0.194607
$539$	−4.06537	−0.175108
$540$	0	0
$541$	28.5501	1.22746	0.613732	−	0.789515i	$-0.289668\pi$
$541$	28.5501	1.22746	0.613732	+	0.789515i	$0.289668\pi$
$542$	15.8496	0.680797
$543$	8.82653	0.378783
$544$	0.387873	0.0166299
$545$	0	0
$546$	−5.40105	−0.231143
$547$	28.4749	1.21750	0.608748	−	0.793363i	$-0.291672\pi$
$547$	28.4749	1.21750	0.608748	+	0.793363i	$0.291672\pi$
$548$	18.1622	0.775851
$549$	11.9248	0.508937
$550$	0	0
$551$	6.96239	0.296608
$552$	−0.962389	−0.0409620
$553$	−11.2243	−0.477304
$554$	−10.3127	−0.438143
$555$	0	0
$556$	8.77575	0.372175
$557$	4.88717	0.207076	0.103538	−	0.994626i	$-0.466984\pi$
$557$	4.88717	0.207076	0.103538	+	0.994626i	$0.466984\pi$
$558$	3.35026	0.141828
$559$	−9.97365	−0.421841
$560$	0	0
$561$	0.373285	0.0157601
$562$	24.3488	1.02709
$563$	−30.8021	−1.29815	−0.649077	−	0.760723i	$-0.724845\pi$
$563$	−30.8021	−1.29815	−0.649077	+	0.760723i	$0.724845\pi$
$564$	0.962389	0.0405239
$565$	0	0
$566$	−26.2882	−1.10498
$567$	3.35026	0.140698
$568$	7.22425	0.303123
$569$	−24.1260	−1.01141	−0.505707	−	0.862705i	$-0.668768\pi$
$569$	−24.1260	−1.01141	−0.505707	+	0.862705i	$0.668768\pi$
$570$	0	0
$571$	−5.67276	−0.237398	−0.118699	−	0.992930i	$-0.537872\pi$
$571$	−5.67276	−0.237398	−0.118699	+	0.992930i	$0.537872\pi$
$572$	−1.55149	−0.0648712
$573$	−2.31265	−0.0966124
$574$	−31.0738	−1.29700
$575$	0	0
$576$	1.00000	0.0416667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	−16.8496	−0.700849
$579$	7.58769	0.315334
$580$	0	0
$581$	−50.2981	−2.08672
$582$	−10.9624	−0.454406
$583$	5.77433	0.239148
$584$	3.22425	0.133421
$585$	0	0
$586$	13.0738	0.540074
$587$	13.6121	0.561833	0.280916	−	0.959732i	$-0.409362\pi$
$587$	13.6121	0.561833	0.280916	+	0.959732i	$0.409362\pi$
$588$	−4.22425	−0.174205
$589$	3.35026	0.138045
$590$	0	0
$591$	−18.8119	−0.773820
$592$	1.61213	0.0662580
$593$	−23.4617	−0.963456	−0.481728	−	0.876321i	$-0.659991\pi$
$593$	−23.4617	−0.963456	−0.481728	+	0.876321i	$0.659991\pi$
$594$	0.962389	0.0394873
$595$	0	0
$596$	15.9756	0.654385
$597$	9.40105	0.384759
$598$	1.55149	0.0634452
$599$	10.0263	0.409665	0.204833	−	0.978797i	$-0.434335\pi$
$599$	10.0263	0.409665	0.204833	+	0.978797i	$0.434335\pi$
$600$	0	0
$601$	11.7743	0.480285	0.240143	−	0.970738i	$-0.422806\pi$
$601$	11.7743	0.480285	0.240143	+	0.970738i	$0.422806\pi$
$602$	−20.7269	−0.844764
$603$	7.22425	0.294194
$604$	18.4241	0.749665
$605$	0	0
$606$	−2.72496	−0.110694
$607$	−38.4993	−1.56264	−0.781319	−	0.624132i	$-0.785453\pi$
$607$	−38.4993	−1.56264	−0.781319	+	0.624132i	$0.785453\pi$
$608$	1.00000	0.0405554
$609$	−23.3258	−0.945210
$610$	0	0
$611$	−1.55149	−0.0627667
$612$	0.387873	0.0156788
$613$	7.61213	0.307451	0.153725	−	0.988114i	$-0.450873\pi$
$613$	7.61213	0.307451	0.153725	+	0.988114i	$0.450873\pi$
$614$	7.07381	0.285476
$615$	0	0
$616$	−3.22425	−0.129909
$617$	29.5369	1.18911	0.594555	−	0.804055i	$-0.297328\pi$
$617$	29.5369	1.18911	0.594555	+	0.804055i	$0.297328\pi$
$618$	0.574515	0.0231104
$619$	−22.5501	−0.906364	−0.453182	−	0.891418i	$-0.649711\pi$
$619$	−22.5501	−0.906364	−0.453182	+	0.891418i	$0.649711\pi$
$620$	0	0
$621$	−0.962389	−0.0386193
$622$	26.3127	1.05504
$623$	15.5778	0.624113
$624$	−1.61213	−0.0645367
$625$	0	0
$626$	18.7005	0.747423
$627$	0.962389	0.0384341
$628$	13.7889	0.550238
$629$	0.625301	0.0249324
$630$	0	0
$631$	−37.9248	−1.50976	−0.754881	−	0.655862i	$-0.772305\pi$
$631$	−37.9248	−1.50976	−0.754881	+	0.655862i	$0.772305\pi$
$632$	−3.35026	−0.133266
$633$	−4.77575	−0.189819
$634$	−26.4749	−1.05145
$635$	0	0
$636$	6.00000	0.237915
$637$	6.81003	0.269823
$638$	−6.70052	−0.265276
$639$	7.22425	0.285787
$640$	0	0
$641$	6.67609	0.263690	0.131845	−	0.991270i	$-0.457910\pi$
$641$	6.67609	0.263690	0.131845	+	0.991270i	$0.457910\pi$
$642$	10.7005	0.422316
$643$	25.2605	0.996175	0.498087	−	0.867127i	$-0.334036\pi$
$643$	25.2605	0.996175	0.498087	+	0.867127i	$0.334036\pi$
$644$	3.22425	0.127053
$645$	0	0
$646$	0.387873	0.0152607
$647$	16.9135	0.664939	0.332469	−	0.943114i	$-0.392118\pi$
$647$	16.9135	0.664939	0.332469	+	0.943114i	$0.392118\pi$
$648$	1.00000	0.0392837
$649$	−9.92478	−0.389582
$650$	0	0
$651$	−11.2243	−0.439913
$652$	3.73813	0.146397
$653$	24.1114	0.943553	0.471776	−	0.881718i	$-0.343613\pi$
$653$	24.1114	0.943553	0.471776	+	0.881718i	$0.343613\pi$
$654$	−10.1260	−0.395958
$655$	0	0
$656$	−9.27504	−0.362129
$657$	3.22425	0.125790
$658$	−3.22425	−0.125694
$659$	−20.3879	−0.794199	−0.397099	−	0.917776i	$-0.629983\pi$
$659$	−20.3879	−0.794199	−0.397099	+	0.917776i	$0.629983\pi$
$660$	0	0
$661$	17.6023	0.684649	0.342325	−	0.939582i	$-0.388786\pi$
$661$	17.6023	0.684649	0.342325	+	0.939582i	$0.388786\pi$
$662$	−30.7005	−1.19321
$663$	−0.625301	−0.0242847
$664$	−15.0132	−0.582624
$665$	0	0
$666$	1.61213	0.0624686
$667$	6.70052	0.259445
$668$	−15.4763	−0.598795
$669$	−24.6761	−0.954033
$670$	0	0
$671$	−11.4763	−0.443036
$672$	−3.35026	−0.129239
$673$	44.3634	1.71008	0.855042	−	0.518558i	$-0.173531\pi$
$673$	44.3634	1.71008	0.855042	+	0.518558i	$0.173531\pi$
$674$	−6.81194	−0.262386
$675$	0	0
$676$	−10.4010	−0.400040
$677$	8.70052	0.334388	0.167194	−	0.985924i	$-0.446529\pi$
$677$	8.70052	0.334388	0.167194	+	0.985924i	$0.446529\pi$
$678$	20.5501	0.789221
$679$	36.7269	1.40945
$680$	0	0
$681$	15.4763	0.593052
$682$	−3.22425	−0.123463
$683$	−37.8759	−1.44928	−0.724641	−	0.689127i	$-0.757994\pi$
$683$	−37.8759	−1.44928	−0.724641	+	0.689127i	$0.757994\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−9.29948	−0.355056
$687$	−21.3258	−0.813631
$688$	−6.18664	−0.235864
$689$	−9.67276	−0.368503
$690$	0	0
$691$	−0.775746	−0.0295108	−0.0147554	−	0.999891i	$-0.504697\pi$
$691$	−0.775746	−0.0295108	−0.0147554	+	0.999891i	$0.504697\pi$
$692$	1.47627	0.0561194
$693$	−3.22425	−0.122479
$694$	−18.3879	−0.697994
$695$	0	0
$696$	−6.96239	−0.263909
$697$	−3.59754	−0.136266
$698$	31.1490	1.17901
$699$	−9.01317	−0.340910
$700$	0	0
$701$	42.3752	1.60049	0.800245	−	0.599674i	$-0.204703\pi$
$701$	42.3752	1.60049	0.800245	+	0.599674i	$0.204703\pi$
$702$	−1.61213	−0.0608458
$703$	1.61213	0.0608025
$704$	−0.962389	−0.0362714
$705$	0	0
$706$	−7.61213	−0.286486
$707$	9.12933	0.343344
$708$	−10.3127	−0.387573
$709$	36.2784	1.36246	0.681231	−	0.732068i	$-0.261445\pi$
$709$	36.2784	1.36246	0.681231	+	0.732068i	$0.261445\pi$
$710$	0	0
$711$	−3.35026	−0.125645
$712$	4.64974	0.174256
$713$	3.22425	0.120749
$714$	−1.29948	−0.0486317
$715$	0	0
$716$	−14.3127	−0.534889
$717$	0.135857	0.00507368
$718$	−35.3112	−1.31780
$719$	−42.5355	−1.58631	−0.793153	−	0.609022i	$-0.791562\pi$
$719$	−42.5355	−1.58631	−0.793153	+	0.609022i	$0.791562\pi$
$720$	0	0
$721$	−1.92478	−0.0716824
$722$	1.00000	0.0372161
$723$	25.8496	0.961355
$724$	−8.82653	−0.328035
$725$	0	0
$726$	10.0738	0.373874
$727$	−0.378024	−0.0140201	−0.00701007	−	0.999975i	$-0.502231\pi$
$727$	−0.378024	−0.0140201	−0.00701007	+	0.999975i	$0.502231\pi$
$728$	5.40105	0.200176
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.39963	−0.0887536
$732$	−11.9248	−0.440752
$733$	26.0118	0.960766	0.480383	−	0.877059i	$-0.340498\pi$
$733$	26.0118	0.960766	0.480383	+	0.877059i	$0.340498\pi$
$734$	−31.9756	−1.18024
$735$	0	0
$736$	0.962389	0.0354741
$737$	−6.95254	−0.256100
$738$	−9.27504	−0.341419
$739$	−44.8773	−1.65084	−0.825419	−	0.564520i	$-0.809061\pi$
$739$	−44.8773	−1.65084	−0.825419	+	0.564520i	$0.809061\pi$
$740$	0	0
$741$	−1.61213	−0.0592230
$742$	−20.1016	−0.737952
$743$	−4.67418	−0.171479	−0.0857394	−	0.996318i	$-0.527325\pi$
$743$	−4.67418	−0.171479	−0.0857394	+	0.996318i	$0.527325\pi$
$744$	−3.35026	−0.122827
$745$	0	0
$746$	−26.4894	−0.969847
$747$	−15.0132	−0.549303
$748$	−0.373285	−0.0136486
$749$	−35.8496	−1.30991
$750$	0	0
$751$	−6.57452	−0.239907	−0.119954	−	0.992779i	$-0.538275\pi$
$751$	−6.57452	−0.239907	−0.119954	+	0.992779i	$0.538275\pi$
$752$	−0.962389	−0.0350947
$753$	10.1114	0.368481
$754$	11.2243	0.408763
$755$	0	0
$756$	−3.35026	−0.121848
$757$	−15.5633	−0.565656	−0.282828	−	0.959171i	$-0.591273\pi$
$757$	−15.5633	−0.565656	−0.282828	+	0.959171i	$0.591273\pi$
$758$	−19.3258	−0.701946
$759$	0.926192	0.0336186
$760$	0	0
$761$	−23.8759	−0.865501	−0.432750	−	0.901514i	$-0.642457\pi$
$761$	−23.8759	−0.865501	−0.432750	+	0.901514i	$0.642457\pi$
$762$	1.35026	0.0489148
$763$	33.9248	1.22816
$764$	2.31265	0.0836688
$765$	0	0
$766$	−3.37470	−0.121933
$767$	16.6253	0.600305
$768$	−1.00000	−0.0360844
$769$	30.4749	1.09895	0.549476	−	0.835510i	$-0.314827\pi$
$769$	30.4749	1.09895	0.549476	+	0.835510i	$0.314827\pi$
$770$	0	0
$771$	26.9986	0.972330
$772$	−7.58769	−0.273087
$773$	16.3272	0.587250	0.293625	−	0.955921i	$-0.405138\pi$
$773$	16.3272	0.587250	0.293625	+	0.955921i	$0.405138\pi$
$774$	−6.18664	−0.222374
$775$	0	0
$776$	10.9624	0.393527
$777$	−5.40105	−0.193761
$778$	−11.3503	−0.406927
$779$	−9.27504	−0.332313
$780$	0	0
$781$	−6.95254	−0.248781
$782$	0.373285	0.0133486
$783$	−6.96239	−0.248815
$784$	4.22425	0.150866
$785$	0	0
$786$	−12.4387	−0.443672
$787$	30.9525	1.10334	0.551669	−	0.834063i	$-0.313991\pi$
$787$	30.9525	1.10334	0.551669	+	0.834063i	$0.313991\pi$
$788$	18.8119	0.670148
$789$	15.0376	0.535353
$790$	0	0
$791$	−68.8481	−2.44796
$792$	−0.962389	−0.0341970
$793$	19.2243	0.682673
$794$	18.8364	0.668478
$795$	0	0
$796$	−9.40105	−0.333211
$797$	14.8773	0.526982	0.263491	−	0.964662i	$-0.415126\pi$
$797$	14.8773	0.526982	0.263491	+	0.964662i	$0.415126\pi$
$798$	−3.35026	−0.118598
$799$	−0.373285	−0.0132059
$800$	0	0
$801$	4.64974	0.164290
$802$	−4.12601	−0.145694
$803$	−3.10299	−0.109502
$804$	−7.22425	−0.254780
$805$	0	0
$806$	5.40105	0.190244
$807$	4.51388	0.158896
$808$	2.72496	0.0958638
$809$	25.4471	0.894672	0.447336	−	0.894366i	$-0.352373\pi$
$809$	25.4471	0.894672	0.447336	+	0.894366i	$0.352373\pi$
$810$	0	0
$811$	−53.3522	−1.87345	−0.936724	−	0.350069i	$-0.886158\pi$
$811$	−53.3522	−1.87345	−0.936724	+	0.350069i	$0.886158\pi$
$812$	23.3258	0.818576
$813$	−15.8496	−0.555868
$814$	−1.55149	−0.0543798
$815$	0	0
$816$	−0.387873	−0.0135783
$817$	−6.18664	−0.216443
$818$	2.52373	0.0882402
$819$	5.40105	0.188728
$820$	0	0
$821$	−27.9756	−0.976354	−0.488177	−	0.872745i	$-0.662338\pi$
$821$	−27.9756	−0.976354	−0.488177	+	0.872745i	$0.662338\pi$
$822$	−18.1622	−0.633480
$823$	−7.87399	−0.274470	−0.137235	−	0.990539i	$-0.543822\pi$
$823$	−7.87399	−0.274470	−0.137235	+	0.990539i	$0.543822\pi$
$824$	−0.574515	−0.0200142
$825$	0	0
$826$	34.5501	1.20215
$827$	40.1016	1.39447	0.697234	−	0.716843i	$-0.254414\pi$
$827$	40.1016	1.39447	0.697234	+	0.716843i	$0.254414\pi$
$828$	0.962389	0.0334453
$829$	47.2262	1.64023	0.820116	−	0.572197i	$-0.193909\pi$
$829$	47.2262	1.64023	0.820116	+	0.572197i	$0.193909\pi$
$830$	0	0
$831$	10.3127	0.357742
$832$	1.61213	0.0558904
$833$	1.63847	0.0567698
$834$	−8.77575	−0.303879
$835$	0	0
$836$	−0.962389	−0.0332849
$837$	−3.35026	−0.115802
$838$	7.51247	0.259514
$839$	−3.89843	−0.134589	−0.0672944	−	0.997733i	$-0.521437\pi$
$839$	−3.89843	−0.134589	−0.0672944	+	0.997733i	$0.521437\pi$
$840$	0	0
$841$	19.4749	0.671547
$842$	3.67750	0.126735
$843$	−24.3488	−0.838619
$844$	4.77575	0.164388
$845$	0	0
$846$	−0.962389	−0.0330876
$847$	−33.7499	−1.15966
$848$	−6.00000	−0.206041
$849$	26.2882	0.902209
$850$	0	0
$851$	1.55149	0.0531845
$852$	−7.22425	−0.247499
$853$	35.1900	1.20488	0.602441	−	0.798164i	$-0.294195\pi$
$853$	35.1900	1.20488	0.602441	+	0.798164i	$0.294195\pi$
$854$	39.9511	1.36710
$855$	0	0
$856$	−10.7005	−0.365736
$857$	−12.1768	−0.415951	−0.207976	−	0.978134i	$-0.566687\pi$
$857$	−12.1768	−0.415951	−0.207976	+	0.978134i	$0.566687\pi$
$858$	1.55149	0.0529671
$859$	57.3522	1.95683	0.978415	−	0.206648i	$-0.0662554\pi$
$859$	57.3522	1.95683	0.978415	+	0.206648i	$0.0662554\pi$
$860$	0	0
$861$	31.0738	1.05899
$862$	−10.3272	−0.351747
$863$	14.0752	0.479126	0.239563	−	0.970881i	$-0.422996\pi$
$863$	14.0752	0.479126	0.239563	+	0.970881i	$0.422996\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−31.5877	−1.07339
$867$	16.8496	0.572241
$868$	11.2243	0.380976
$869$	3.22425	0.109375
$870$	0	0
$871$	11.6464	0.394624
$872$	10.1260	0.342910
$873$	10.9624	0.371021
$874$	0.962389	0.0325533
$875$	0	0
$876$	−3.22425	−0.108937
$877$	−18.8627	−0.636949	−0.318475	−	0.947931i	$-0.603171\pi$
$877$	−18.8627	−0.636949	−0.318475	+	0.947931i	$0.603171\pi$
$878$	−38.1524	−1.28758
$879$	−13.0738	−0.440969
$880$	0	0
$881$	−19.1490	−0.645147	−0.322574	−	0.946544i	$-0.604548\pi$
$881$	−19.1490	−0.645147	−0.322574	+	0.946544i	$0.604548\pi$
$882$	4.22425	0.142238
$883$	−42.9135	−1.44415	−0.722077	−	0.691812i	$-0.756813\pi$
$883$	−42.9135	−1.44415	−0.722077	+	0.691812i	$0.756813\pi$
$884$	0.625301	0.0210311
$885$	0	0
$886$	−16.3127	−0.548034
$887$	−22.7005	−0.762209	−0.381104	−	0.924532i	$-0.624456\pi$
$887$	−22.7005	−0.762209	−0.381104	+	0.924532i	$0.624456\pi$
$888$	−1.61213	−0.0540994
$889$	−4.52373	−0.151721
$890$	0	0
$891$	−0.962389	−0.0322412
$892$	24.6761	0.826216
$893$	−0.962389	−0.0322051
$894$	−15.9756	−0.534303
$895$	0	0
$896$	3.35026	0.111924
$897$	−1.55149	−0.0518028
$898$	−37.5271	−1.25229
$899$	23.3258	0.777960
$900$	0	0
$901$	−2.32724	−0.0775316
$902$	8.92619	0.297210
$903$	20.7269	0.689747
$904$	−20.5501	−0.683485
$905$	0	0
$906$	−18.4241	−0.612099
$907$	−14.5501	−0.483127	−0.241564	−	0.970385i	$-0.577660\pi$
$907$	−14.5501	−0.483127	−0.241564	+	0.970385i	$0.577660\pi$
$908$	−15.4763	−0.513598
$909$	2.72496	0.0903813
$910$	0	0
$911$	−0.998585	−0.0330846	−0.0165423	−	0.999863i	$-0.505266\pi$
$911$	−0.998585	−0.0330846	−0.0165423	+	0.999863i	$0.505266\pi$
$912$	−1.00000	−0.0331133
$913$	14.4485	0.478176
$914$	8.00000	0.264616
$915$	0	0
$916$	21.3258	0.704625
$917$	41.6728	1.37616
$918$	−0.387873	−0.0128017
$919$	−33.7743	−1.11411	−0.557056	−	0.830475i	$-0.688069\pi$
$919$	−33.7743	−1.11411	−0.557056	+	0.830475i	$0.688069\pi$
$920$	0	0
$921$	−7.07381	−0.233090
$922$	12.3780	0.407649
$923$	11.6464	0.383346
$924$	3.22425	0.106070
$925$	0	0
$926$	−32.4504	−1.06639
$927$	−0.574515	−0.0188696
$928$	6.96239	0.228552
$929$	14.8773	0.488109	0.244054	−	0.969762i	$-0.421522\pi$
$929$	14.8773	0.488109	0.244054	+	0.969762i	$0.421522\pi$
$930$	0	0
$931$	4.22425	0.138444
$932$	9.01317	0.295236
$933$	−26.3127	−0.861437
$934$	−7.53690	−0.246615
$935$	0	0
$936$	1.61213	0.0526940
$937$	−22.2981	−0.728446	−0.364223	−	0.931312i	$-0.618665\pi$
$937$	−22.2981	−0.728446	−0.364223	+	0.931312i	$0.618665\pi$
$938$	24.2031	0.790261
$939$	−18.7005	−0.610269
$940$	0	0
$941$	−36.3634	−1.18541	−0.592707	−	0.805418i	$-0.701941\pi$
$941$	−36.3634	−1.18541	−0.592707	+	0.805418i	$0.701941\pi$
$942$	−13.7889	−0.449267
$943$	−8.92619	−0.290677
$944$	10.3127	0.335648
$945$	0	0
$946$	5.95395	0.193580
$947$	−50.3390	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$947$	−50.3390	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$948$	3.35026	0.108811
$949$	5.19791	0.168731
$950$	0	0
$951$	26.4749	0.858506
$952$	1.29948	0.0421163
$953$	16.3272	0.528891	0.264446	−	0.964401i	$-0.414811\pi$
$953$	16.3272	0.528891	0.264446	+	0.964401i	$0.414811\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−0.135857	−0.00439393
$957$	6.70052	0.216597
$958$	15.2097	0.491402
$959$	60.8481	1.96489
$960$	0	0
$961$	−19.7757	−0.637927
$962$	2.59895	0.0837936
$963$	−10.7005	−0.344820
$964$	−25.8496	−0.832558
$965$	0	0
$966$	−3.22425	−0.103739
$967$	−0.276454	−0.00889015	−0.00444507	−	0.999990i	$-0.501415\pi$
$967$	−0.276454	−0.00889015	−0.00444507	+	0.999990i	$0.501415\pi$
$968$	−10.0738	−0.323784
$969$	−0.387873	−0.0124603
$970$	0	0
$971$	37.9102	1.21660	0.608298	−	0.793709i	$-0.291853\pi$
$971$	37.9102	1.21660	0.608298	+	0.793709i	$0.291853\pi$
$972$	−1.00000	−0.0320750
$973$	29.4010	0.942554
$974$	1.19982	0.0384446
$975$	0	0
$976$	11.9248	0.381703
$977$	−28.3996	−0.908585	−0.454292	−	0.890853i	$-0.650108\pi$
$977$	−28.3996	−0.908585	−0.454292	+	0.890853i	$0.650108\pi$
$978$	−3.73813	−0.119532
$979$	−4.47486	−0.143017
$980$	0	0
$981$	10.1260	0.323299
$982$	−5.11283	−0.163157
$983$	0.926192	0.0295409	0.0147705	−	0.999891i	$-0.495298\pi$
$983$	0.926192	0.0295409	0.0147705	+	0.999891i	$0.495298\pi$
$984$	9.27504	0.295677
$985$	0	0
$986$	2.70052	0.0860022
$987$	3.22425	0.102629
$988$	1.61213	0.0512886
$989$	−5.95395	−0.189325
$990$	0	0
$991$	30.5256	0.969679	0.484839	−	0.874603i	$-0.338878\pi$
$991$	30.5256	0.969679	0.484839	+	0.874603i	$0.338878\pi$
$992$	3.35026	0.106371
$993$	30.7005	0.974252
$994$	24.2031	0.767677
$995$	0	0
$996$	15.0132	0.475711
$997$	−21.0132	−0.665494	−0.332747	−	0.943016i	$-0.607975\pi$
$997$	−21.0132	−0.665494	−0.332747	+	0.943016i	$0.607975\pi$
$998$	14.2981	0.452597
$999$	−1.61213	−0.0510054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.bm.1.3		3
3.2	odd	2		8550.2.a.ce.1.3		3
5.2	odd	4		570.2.d.c.229.4	yes	6
5.3	odd	4		570.2.d.c.229.1	✓	6
5.4	even	2		2850.2.a.bl.1.1		3
15.2	even	4		1710.2.d.f.1369.3		6
15.8	even	4		1710.2.d.f.1369.6		6
15.14	odd	2		8550.2.a.cq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.c.229.1	✓	6	5.3	odd	4
570.2.d.c.229.4	yes	6	5.2	odd	4
1710.2.d.f.1369.3		6	15.2	even	4
1710.2.d.f.1369.6		6	15.8	even	4
2850.2.a.bl.1.1		3	5.4	even	2
2850.2.a.bm.1.3		3	1.1	even	1	trivial
8550.2.a.ce.1.3		3	3.2	odd	2
8550.2.a.cq.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.35026 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.962389 q^{11} -1.00000 q^{12} +1.61213 q^{13} +3.35026 q^{14} +1.00000 q^{16} +0.387873 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.35026 q^{21} -0.962389 q^{22} +0.962389 q^{23} -1.00000 q^{24} +1.61213 q^{26} -1.00000 q^{27} +3.35026 q^{28} +6.96239 q^{29} +3.35026 q^{31} +1.00000 q^{32} +0.962389 q^{33} +0.387873 q^{34} +1.00000 q^{36} +1.61213 q^{37} +1.00000 q^{38} -1.61213 q^{39} -9.27504 q^{41} -3.35026 q^{42} -6.18664 q^{43} -0.962389 q^{44} +0.962389 q^{46} -0.962389 q^{47} -1.00000 q^{48} +4.22425 q^{49} -0.387873 q^{51} +1.61213 q^{52} -6.00000 q^{53} -1.00000 q^{54} +3.35026 q^{56} -1.00000 q^{57} +6.96239 q^{58} +10.3127 q^{59} +11.9248 q^{61} +3.35026 q^{62} +3.35026 q^{63} +1.00000 q^{64} +0.962389 q^{66} +7.22425 q^{67} +0.387873 q^{68} -0.962389 q^{69} +7.22425 q^{71} +1.00000 q^{72} +3.22425 q^{73} +1.61213 q^{74} +1.00000 q^{76} -3.22425 q^{77} -1.61213 q^{78} -3.35026 q^{79} +1.00000 q^{81} -9.27504 q^{82} -15.0132 q^{83} -3.35026 q^{84} -6.18664 q^{86} -6.96239 q^{87} -0.962389 q^{88} +4.64974 q^{89} +5.40105 q^{91} +0.962389 q^{92} -3.35026 q^{93} -0.962389 q^{94} -1.00000 q^{96} +10.9624 q^{97} +4.22425 q^{98} -0.962389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 8 q^{11} - 3 q^{12} + 4 q^{13} + 3 q^{16} + 2 q^{17} + 3 q^{18} + 3 q^{19} + 8 q^{22} - 8 q^{23} - 3 q^{24} + 4 q^{26} - 3 q^{27} + 10 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 8 * q^11 - 3 * q^12 + 4 * q^13 + 3 * q^16 + 2 * q^17 + 3 * q^18 + 3 * q^19 + 8 * q^22 - 8 * q^23 - 3 * q^24 + 4 * q^26 - 3 * q^27 + 10 * q^29 + 3 * q^32 - 8 * q^33 + 2 * q^34 + 3 * q^36 + 4 * q^37 + 3 * q^38 - 4 * q^39 + 4 * q^41 - 6 * q^43 + 8 * q^44 - 8 * q^46 + 8 * q^47 - 3 * q^48 + 11 * q^49 - 2 * q^51 + 4 * q^52 - 18 * q^53 - 3 * q^54 - 3 * q^57 + 10 * q^58 + 10 * q^59 + 14 * q^61 + 3 * q^64 - 8 * q^66 + 20 * q^67 + 2 * q^68 + 8 * q^69 + 20 * q^71 + 3 * q^72 + 8 * q^73 + 4 * q^74 + 3 * q^76 - 8 * q^77 - 4 * q^78 + 3 * q^81 + 4 * q^82 - 4 * q^83 - 6 * q^86 - 10 * q^87 + 8 * q^88 + 24 * q^89 - 24 * q^91 - 8 * q^92 + 8 * q^94 - 3 * q^96 + 22 * q^97 + 11 * q^98 + 8 * q^99

Introduction

L-functions

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Database

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