LMFDB - Embedded newform 1850.2.b.o.149.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1850,2,Mod(149,1850)] 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("1850.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,0,0,0,-22,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3182656.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 25x^{2} - 10x + 2 $ x^6 - 2x^3 + 25x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.4
Root			$1.46962 + 1.46962i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.o.149.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -2.93923i q^{3} -1.00000 q^{4} +2.93923 q^{6} -1.31955i q^{7} -1.00000i q^{8} -5.63910 q^{9} -0.258786 q^{11} +2.93923i q^{12} +5.87847i q^{13} +1.31955 q^{14} +1.00000 q^{16} +4.25879i q^{17} -5.63910i q^{18} -2.93923 q^{19} -3.87847 q^{21} -0.258786i q^{22} +8.51757i q^{23} -2.93923 q^{24} -5.87847 q^{26} +7.75694i q^{27} +1.31955i q^{28} +3.61968 q^{29} +3.95865 q^{31} +1.00000i q^{32} +0.760632i q^{33} -4.25879 q^{34} +5.63910 q^{36} -1.00000i q^{37} -2.93923i q^{38} +17.2782 q^{39} -8.89789 q^{41} -3.87847i q^{42} +5.61968i q^{43} +0.258786 q^{44} -8.51757 q^{46} +5.57834i q^{47} -2.93923i q^{48} +5.25879 q^{49} +12.5176 q^{51} -5.87847i q^{52} -12.8979i q^{53} -7.75694 q^{54} -1.31955 q^{56} +8.63910i q^{57} +3.61968i q^{58} -9.57834 q^{59} +0.380316 q^{61} +3.95865i q^{62} +7.44108i q^{63} -1.00000 q^{64} -0.760632 q^{66} +5.57834i q^{67} -4.25879i q^{68} +25.0351 q^{69} +11.2394 q^{71} +5.63910i q^{72} -2.51757i q^{73} +1.00000 q^{74} +2.93923 q^{76} +0.341481i q^{77} +17.2782i q^{78} +7.69987 q^{79} +5.88216 q^{81} -8.89789i q^{82} +6.17860i q^{83} +3.87847 q^{84} -5.61968 q^{86} -10.6391i q^{87} +0.258786i q^{88} -6.00000 q^{89} +7.75694 q^{91} -8.51757i q^{92} -11.6354i q^{93} -5.57834 q^{94} +2.93923 q^{96} -12.8979i q^{97} +5.25879i q^{98} +1.45932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56}+ \cdots - 38 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 - 22 * q^9 + 22 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^21 + 10 * q^29 + 6 * q^31 - 2 * q^34 + 22 * q^36 + 80 * q^39 - 18 * q^41 - 22 * q^44 - 4 * q^46 + 8 * q^49 + 28 * q^51 + 24 * q^54 - 2 * q^56 - 28 * q^59 + 14 * q^61 - 6 * q^64 - 28 * q^66 + 56 * q^69 + 44 * q^71 + 6 * q^74 + 52 * q^79 + 94 * q^81 - 12 * q^84 - 22 * q^86 - 36 * q^89 - 24 * q^91 - 4 * q^94 - 38 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 2.93923i	− 1.69697i	−0.529221	−	0.848484i	$-0.677516\pi$
$3$	− 2.93923i	− 1.69697i	0.529221	−	0.848484i	$-0.322484\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.93923	1.19994
$7$	− 1.31955i	− 0.498743i	−0.968408	−	0.249372i	$-0.919776\pi$
$7$	− 1.31955i	− 0.498743i	0.968408	−	0.249372i	$-0.0802241\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−5.63910	−1.87970
$10$	0	0
$11$	−0.258786	−0.0780268	−0.0390134	−	0.999239i	$-0.512422\pi$
$11$	−0.258786	−0.0780268	−0.0390134	+	0.999239i	$0.512422\pi$
$12$	2.93923i	0.848484i
$13$	5.87847i	1.63039i	0.579184	+	0.815197i	$0.303371\pi$
$13$	5.87847i	1.63039i	−0.579184	+	0.815197i	$0.696629\pi$
$14$	1.31955	0.352665
$15$	0	0
$16$	1.00000	0.250000
$17$	4.25879i	1.03291i	0.856315	+	0.516454i	$0.172748\pi$
$17$	4.25879i	1.03291i	−0.856315	+	0.516454i	$0.827252\pi$
$18$	− 5.63910i	− 1.32915i
$19$	−2.93923	−0.674307	−0.337153	−	0.941450i	$-0.609464\pi$
$19$	−2.93923	−0.674307	−0.337153	+	0.941450i	$0.609464\pi$
$20$	0	0
$21$	−3.87847	−0.846351
$22$	− 0.258786i	− 0.0551733i
$23$	8.51757i	1.77604i	0.459808	+	0.888018i	$0.347918\pi$
$23$	8.51757i	1.77604i	−0.459808	+	0.888018i	$0.652082\pi$
$24$	−2.93923	−0.599969
$25$	0	0
$26$	−5.87847	−1.15286
$27$	7.75694i	1.49282i
$28$	1.31955i	0.249372i
$29$	3.61968	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$29$	3.61968	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$30$	0	0
$31$	3.95865	0.710995	0.355497	−	0.934677i	$-0.384311\pi$
$31$	3.95865	0.710995	0.355497	+	0.934677i	$0.384311\pi$
$32$	1.00000i	0.176777i
$33$	0.760632i	0.132409i
$34$	−4.25879	−0.730376
$35$	0	0
$36$	5.63910	0.939850
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	− 2.93923i	− 0.476807i
$39$	17.2782	2.76673
$40$	0	0
$41$	−8.89789	−1.38962	−0.694808	−	0.719195i	$-0.744511\pi$
$41$	−8.89789	−1.38962	−0.694808	+	0.719195i	$0.744511\pi$
$42$	− 3.87847i	− 0.598461i
$43$	5.61968i	0.856994i	0.903543	+	0.428497i	$0.140957\pi$
$43$	5.61968i	0.856994i	−0.903543	+	0.428497i	$0.859043\pi$
$44$	0.258786	0.0390134
$45$	0	0
$46$	−8.51757	−1.25585
$47$	5.57834i	0.813684i	0.913499	+	0.406842i	$0.133370\pi$
$47$	5.57834i	0.813684i	−0.913499	+	0.406842i	$0.866630\pi$
$48$	− 2.93923i	− 0.424242i
$49$	5.25879	0.751255
$50$	0	0
$51$	12.5176	1.75281
$52$	− 5.87847i	− 0.815197i
$53$	− 12.8979i	− 1.77166i	−0.464009	−	0.885831i	$-0.653589\pi$
$53$	− 12.8979i	− 1.77166i	0.464009	−	0.885831i	$-0.346411\pi$
$54$	−7.75694	−1.05559
$55$	0	0
$56$	−1.31955	−0.176332
$57$	8.63910i	1.14428i
$58$	3.61968i	0.475288i
$59$	−9.57834	−1.24699	−0.623497	−	0.781826i	$-0.714288\pi$
$59$	−9.57834	−1.24699	−0.623497	+	0.781826i	$0.714288\pi$
$60$	0	0
$61$	0.380316	0.0486945	0.0243472	−	0.999704i	$-0.492249\pi$
$61$	0.380316	0.0486945	0.0243472	+	0.999704i	$0.492249\pi$
$62$	3.95865i	0.502749i
$63$	7.44108i	0.937488i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−0.760632	−0.0936273
$67$	5.57834i	0.681502i	0.940154	+	0.340751i	$0.110681\pi$
$67$	5.57834i	0.681502i	−0.940154	+	0.340751i	$0.889319\pi$
$68$	− 4.25879i	− 0.516454i
$69$	25.0351	3.01388
$70$	0	0
$71$	11.2394	1.33387	0.666934	−	0.745117i	$-0.267606\pi$
$71$	11.2394	1.33387	0.666934	+	0.745117i	$0.267606\pi$
$72$	5.63910i	0.664574i
$73$	− 2.51757i	− 0.294659i	−0.989087	−	0.147330i	$-0.952932\pi$
$73$	− 2.51757i	− 0.294659i	0.989087	−	0.147330i	$-0.0470678\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.93923	0.337153
$77$	0.341481i	0.0389154i
$78$	17.2782i	1.95637i
$79$	7.69987	0.866303	0.433151	−	0.901321i	$-0.357402\pi$
$79$	7.69987	0.866303	0.433151	+	0.901321i	$0.357402\pi$
$80$	0	0
$81$	5.88216	0.653574
$82$	− 8.89789i	− 0.982607i
$83$	6.17860i	0.678190i	0.940752	+	0.339095i	$0.110121\pi$
$83$	6.17860i	0.678190i	−0.940752	+	0.339095i	$0.889879\pi$
$84$	3.87847	0.423176
$85$	0	0
$86$	−5.61968	−0.605986
$87$	− 10.6391i	− 1.14063i
$88$	0.258786i	0.0275866i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	7.75694	0.813148
$92$	− 8.51757i	− 0.888018i
$93$	− 11.6354i	− 1.20654i
$94$	−5.57834	−0.575361
$95$	0	0
$96$	2.93923	0.299984
$97$	− 12.8979i	− 1.30958i	−0.755810	−	0.654791i	$-0.772757\pi$
$97$	− 12.8979i	− 1.30958i	0.755810	−	0.654791i	$-0.227243\pi$
$98$	5.25879i	0.531218i
$99$	1.45932	0.146667
$100$	0	0
$101$	10.3960	1.03444	0.517222	−	0.855851i	$-0.326966\pi$
$101$	10.3960	1.03444	0.517222	+	0.855851i	$0.326966\pi$
$102$	12.5176i	1.23942i
$103$	13.8785i	1.36749i	0.729723	+	0.683743i	$0.239649\pi$
$103$	13.8785i	1.36749i	−0.729723	+	0.683743i	$0.760351\pi$
$104$	5.87847	0.576431
$105$	0	0
$106$	12.8979	1.25275
$107$	− 4.21744i	− 0.407715i	−0.979001	−	0.203858i	$-0.934652\pi$
$107$	− 4.21744i	− 0.407715i	0.979001	−	0.203858i	$-0.0653479\pi$
$108$	− 7.75694i	− 0.746412i
$109$	−16.1373	−1.54567	−0.772834	−	0.634608i	$-0.781162\pi$
$109$	−16.1373	−1.54567	−0.772834	+	0.634608i	$0.781162\pi$
$110$	0	0
$111$	−2.93923	−0.278980
$112$	− 1.31955i	− 0.124686i
$113$	− 1.01942i	− 0.0958987i	−0.998850	−	0.0479494i	$-0.984731\pi$
$113$	− 1.01942i	− 0.0958987i	0.998850	−	0.0479494i	$-0.0152686\pi$
$114$	−8.63910	−0.809126
$115$	0	0
$116$	−3.61968	−0.336079
$117$	− 33.1493i	− 3.06465i
$118$	− 9.57834i	− 0.881757i
$119$	5.61968	0.515156
$120$	0	0
$121$	−10.9330	−0.993912
$122$	0.380316i	0.0344322i
$123$	26.1530i	2.35813i
$124$	−3.95865	−0.355497
$125$	0	0
$126$	−7.44108	−0.662904
$127$	7.45681i	0.661685i	0.943686	+	0.330842i	$0.107333\pi$
$127$	7.45681i	0.661685i	−0.943686	+	0.330842i	$0.892667\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	16.5176	1.45429
$130$	0	0
$131$	2.42166	0.211582	0.105791	−	0.994388i	$-0.466263\pi$
$131$	2.42166	0.211582	0.105791	+	0.994388i	$0.466263\pi$
$132$	− 0.760632i	− 0.0662045i
$133$	3.87847i	0.336306i
$134$	−5.57834	−0.481895
$135$	0	0
$136$	4.25879	0.365188
$137$	15.7958i	1.34952i	0.738035	+	0.674762i	$0.235754\pi$
$137$	15.7958i	1.34952i	−0.738035	+	0.674762i	$0.764246\pi$
$138$	25.0351i	2.13113i
$139$	11.4155	0.968247	0.484123	−	0.875000i	$-0.339139\pi$
$139$	11.4155	0.968247	0.484123	+	0.875000i	$0.339139\pi$
$140$	0	0
$141$	16.3960	1.38080
$142$	11.2394i	0.943187i
$143$	− 1.52126i	− 0.127214i
$144$	−5.63910	−0.469925
$145$	0	0
$146$	2.51757	0.208356
$147$	− 15.4568i	− 1.27486i
$148$	1.00000i	0.0821995i
$149$	−18.3960	−1.50706	−0.753531	−	0.657412i	$-0.771651\pi$
$149$	−18.3960	−1.50706	−0.753531	+	0.657412i	$0.771651\pi$
$150$	0	0
$151$	5.87847	0.478383	0.239192	−	0.970972i	$-0.423118\pi$
$151$	5.87847	0.478383	0.239192	+	0.970972i	$0.423118\pi$
$152$	2.93923i	0.238403i
$153$	− 24.0157i	− 1.94156i
$154$	−0.341481	−0.0275173
$155$	0	0
$156$	−17.2782	−1.38336
$157$	3.61968i	0.288882i	0.989513	+	0.144441i	$0.0461384\pi$
$157$	3.61968i	0.288882i	−0.989513	+	0.144441i	$0.953862\pi$
$158$	7.69987i	0.612569i
$159$	−37.9099	−3.00645
$160$	0	0
$161$	11.2394	0.885786
$162$	5.88216i	0.462146i
$163$	22.0546i	1.72745i	0.503966	+	0.863723i	$0.331874\pi$
$163$	22.0546i	1.72745i	−0.503966	+	0.863723i	$0.668126\pi$
$164$	8.89789	0.694808
$165$	0	0
$166$	−6.17860	−0.479553
$167$	− 0.600267i	− 0.0464500i	−0.999730	−	0.0232250i	$-0.992607\pi$
$167$	− 0.600267i	− 0.0464500i	0.999730	−	0.0232250i	$-0.00739341\pi$
$168$	3.87847i	0.299230i
$169$	−21.5564	−1.65819
$170$	0	0
$171$	16.5746	1.26749
$172$	− 5.61968i	− 0.428497i
$173$	22.6937i	1.72537i	0.505744	+	0.862684i	$0.331218\pi$
$173$	22.6937i	1.72537i	−0.505744	+	0.862684i	$0.668782\pi$
$174$	10.6391	0.806548
$175$	0	0
$176$	−0.258786	−0.0195067
$177$	28.1530i	2.11611i
$178$	− 6.00000i	− 0.449719i
$179$	8.73501	0.652885	0.326443	−	0.945217i	$-0.394150\pi$
$179$	8.73501	0.652885	0.326443	+	0.945217i	$0.394150\pi$
$180$	0	0
$181$	10.6391	0.790798	0.395399	−	0.918509i	$-0.370606\pi$
$181$	10.6391	0.790798	0.395399	+	0.918509i	$0.370606\pi$
$182$	7.75694i	0.574983i
$183$	− 1.11784i	− 0.0826330i
$184$	8.51757	0.627924
$185$	0	0
$186$	11.6354	0.853150
$187$	− 1.10211i	− 0.0805945i
$188$	− 5.57834i	− 0.406842i
$189$	10.2357	0.744536
$190$	0	0
$191$	22.1116	1.59994	0.799971	−	0.600039i	$-0.204848\pi$
$191$	22.1116	1.59994	0.799971	+	0.600039i	$0.204848\pi$
$192$	2.93923i	0.212121i
$193$	− 7.03514i	− 0.506401i	−0.967414	−	0.253200i	$-0.918517\pi$
$193$	− 7.03514i	− 0.506401i	0.967414	−	0.253200i	$-0.0814832\pi$
$194$	12.8979	0.926014
$195$	0	0
$196$	−5.25879	−0.375628
$197$	25.7569i	1.83511i	0.397614	+	0.917553i	$0.369838\pi$
$197$	25.7569i	1.83511i	−0.397614	+	0.917553i	$0.630162\pi$
$198$	1.45932i	0.103709i
$199$	12.7350	0.902761	0.451380	−	0.892332i	$-0.350932\pi$
$199$	12.7350	0.902761	0.451380	+	0.892332i	$0.350932\pi$
$200$	0	0
$201$	16.3960	1.15649
$202$	10.3960i	0.731463i
$203$	− 4.77636i	− 0.335235i
$204$	−12.5176	−0.876405
$205$	0	0
$206$	−13.8785	−0.966959
$207$	− 48.0315i	− 3.33842i
$208$	5.87847i	0.407599i
$209$	0.760632	0.0526140
$210$	0	0
$211$	−6.65483	−0.458137	−0.229069	−	0.973410i	$-0.573568\pi$
$211$	−6.65483	−0.458137	−0.229069	+	0.973410i	$0.573568\pi$
$212$	12.8979i	0.885831i
$213$	− 33.0351i	− 2.26353i
$214$	4.21744	0.288298
$215$	0	0
$216$	7.75694	0.527793
$217$	− 5.22364i	− 0.354604i
$218$	− 16.1373i	− 1.09295i
$219$	−7.39973	−0.500028
$220$	0	0
$221$	−25.0351	−1.68405
$222$	− 2.93923i	− 0.197269i
$223$	16.5589i	1.10887i	0.832228	+	0.554434i	$0.187065\pi$
$223$	16.5589i	1.10887i	−0.832228	+	0.554434i	$0.812935\pi$
$224$	1.31955	0.0881662
$225$	0	0
$226$	1.01942	0.0678106
$227$	− 4.01573i	− 0.266533i	−0.991080	−	0.133267i	$-0.957453\pi$
$227$	− 4.01573i	− 0.266533i	0.991080	−	0.133267i	$-0.0425466\pi$
$228$	− 8.63910i	− 0.572138i
$229$	−25.5139	−1.68600	−0.843002	−	0.537910i	$-0.819214\pi$
$229$	−25.5139	−1.68600	−0.843002	+	0.537910i	$0.819214\pi$
$230$	0	0
$231$	1.00369	0.0660381
$232$	− 3.61968i	− 0.237644i
$233$	− 7.96116i	− 0.521553i	−0.965399	−	0.260777i	$-0.916021\pi$
$233$	− 7.96116i	− 0.521553i	0.965399	−	0.260777i	$-0.0839786\pi$
$234$	33.1493	2.16704
$235$	0	0
$236$	9.57834	0.623497
$237$	− 22.6317i	− 1.47009i
$238$	5.61968i	0.364270i
$239$	−11.7983	−0.763168	−0.381584	−	0.924334i	$-0.624621\pi$
$239$	−11.7983	−0.763168	−0.381584	+	0.924334i	$0.624621\pi$
$240$	0	0
$241$	24.9963	1.61015	0.805077	−	0.593171i	$-0.202124\pi$
$241$	24.9963	1.61015	0.805077	+	0.593171i	$0.202124\pi$
$242$	− 10.9330i	− 0.702802i
$243$	5.98176i	0.383730i
$244$	−0.380316	−0.0243472
$245$	0	0
$246$	−26.1530	−1.66745
$247$	− 17.2782i	− 1.09939i
$248$	− 3.95865i	− 0.251375i
$249$	18.1604	1.15087
$250$	0	0
$251$	−8.81770	−0.556569	−0.278284	−	0.960499i	$-0.589766\pi$
$251$	−8.81770	−0.556569	−0.278284	+	0.960499i	$0.589766\pi$
$252$	− 7.44108i	− 0.468744i
$253$	− 2.20423i	− 0.138578i
$254$	−7.45681	−0.467882
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.8396i	0.738536i	0.929323	+	0.369268i	$0.120392\pi$
$257$	11.8396i	0.738536i	−0.929323	+	0.369268i	$0.879608\pi$
$258$	16.5176i	1.02834i
$259$	−1.31955	−0.0819929
$260$	0	0
$261$	−20.4118	−1.26346
$262$	2.42166i	0.149611i
$263$	9.15919i	0.564780i	0.959300	+	0.282390i	$0.0911271\pi$
$263$	9.15919i	0.564780i	−0.959300	+	0.282390i	$0.908873\pi$
$264$	0.760632	0.0468137
$265$	0	0
$266$	−3.87847	−0.237804
$267$	17.6354i	1.07927i
$268$	− 5.57834i	− 0.340751i
$269$	−18.3960	−1.12163	−0.560813	−	0.827942i	$-0.689511\pi$
$269$	−18.3960	−1.12163	−0.560813	+	0.827942i	$0.689511\pi$
$270$	0	0
$271$	−27.2394	−1.65467	−0.827337	−	0.561706i	$-0.810145\pi$
$271$	−27.2394	−1.65467	−0.827337	+	0.561706i	$0.810145\pi$
$272$	4.25879i	0.258227i
$273$	− 22.7995i	− 1.37989i
$274$	−15.7958	−0.954258
$275$	0	0
$276$	−25.0351	−1.50694
$277$	− 21.8785i	− 1.31455i	−0.753651	−	0.657275i	$-0.771709\pi$
$277$	− 21.8785i	− 1.31455i	0.753651	−	0.657275i	$-0.228291\pi$
$278$	11.4155i	0.684654i
$279$	−22.3232	−1.33646
$280$	0	0
$281$	−20.9963	−1.25253	−0.626267	−	0.779608i	$-0.715418\pi$
$281$	−20.9963	−1.25253	−0.626267	+	0.779608i	$0.715418\pi$
$282$	16.3960i	0.976370i
$283$	− 18.9136i	− 1.12430i	−0.827036	−	0.562149i	$-0.809975\pi$
$283$	− 18.9136i	− 1.12430i	0.827036	−	0.562149i	$-0.190025\pi$
$284$	−11.2394	−0.666934
$285$	0	0
$286$	1.52126	0.0899542
$287$	11.7412i	0.693062i
$288$	− 5.63910i	− 0.332287i
$289$	−1.13726	−0.0668974
$290$	0	0
$291$	−37.9099	−2.22232
$292$	2.51757i	0.147330i
$293$	12.8979i	0.753503i	0.926314	+	0.376751i	$0.122959\pi$
$293$	12.8979i	0.753503i	−0.926314	+	0.376751i	$0.877041\pi$
$294$	15.4568	0.901459
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	− 2.00738i	− 0.116480i
$298$	− 18.3960i	− 1.06565i
$299$	−50.0703	−2.89564
$300$	0	0
$301$	7.41546	0.427420
$302$	5.87847i	0.338268i
$303$	− 30.5564i	− 1.75542i
$304$	−2.93923	−0.168577
$305$	0	0
$306$	24.0157	1.37289
$307$	− 3.78256i	− 0.215882i	−0.994157	−	0.107941i	$-0.965574\pi$
$307$	− 3.78256i	− 0.215882i	0.994157	−	0.107941i	$-0.0344258\pi$
$308$	− 0.341481i	− 0.0194577i
$309$	40.7921	2.32058
$310$	0	0
$311$	−15.2807	−0.866490	−0.433245	−	0.901276i	$-0.642631\pi$
$311$	−15.2807	−0.866490	−0.433245	+	0.901276i	$0.642631\pi$
$312$	− 17.2782i	− 0.978186i
$313$	− 19.0351i	− 1.07593i	−0.842967	−	0.537965i	$-0.819193\pi$
$313$	− 19.0351i	− 1.07593i	0.842967	−	0.537965i	$-0.180807\pi$
$314$	−3.61968	−0.204271
$315$	0	0
$316$	−7.69987	−0.433151
$317$	− 24.1373i	− 1.35568i	−0.735208	−	0.677842i	$-0.762915\pi$
$317$	− 24.1373i	− 1.35568i	0.735208	−	0.677842i	$-0.237085\pi$
$318$	− 37.9099i	− 2.12588i
$319$	−0.936722	−0.0524464
$320$	0	0
$321$	−12.3960	−0.691880
$322$	11.2394i	0.626345i
$323$	− 12.5176i	− 0.696496i
$324$	−5.88216	−0.326787
$325$	0	0
$326$	−22.0546	−1.22149
$327$	47.4312i	2.62295i
$328$	8.89789i	0.491304i
$329$	7.36090	0.405819
$330$	0	0
$331$	0.817705	0.0449451	0.0224726	−	0.999747i	$-0.492846\pi$
$331$	0.817705	0.0449451	0.0224726	+	0.999747i	$0.492846\pi$
$332$	− 6.17860i	− 0.339095i
$333$	5.63910i	0.309021i
$334$	0.600267	0.0328451
$335$	0	0
$336$	−3.87847	−0.211588
$337$	− 19.7958i	− 1.07834i	−0.842195	−	0.539172i	$-0.818737\pi$
$337$	− 19.7958i	− 1.07834i	0.842195	−	0.539172i	$-0.181263\pi$
$338$	− 21.5564i	− 1.17251i
$339$	−2.99631	−0.162737
$340$	0	0
$341$	−1.02444	−0.0554767
$342$	16.5746i	0.896254i
$343$	− 16.1761i	− 0.873427i
$344$	5.61968	0.302993
$345$	0	0
$346$	−22.6937	−1.22002
$347$	15.7569i	0.845877i	0.906158	+	0.422938i	$0.139001\pi$
$347$	15.7569i	0.845877i	−0.906158	+	0.422938i	$0.860999\pi$
$348$	10.6391i	0.570316i
$349$	0.160365	0.00858416	0.00429208	−	0.999991i	$-0.498634\pi$
$349$	0.160365	0.00858416	0.00429208	+	0.999991i	$0.498634\pi$
$350$	0	0
$351$	−45.5989	−2.43389
$352$	− 0.258786i	− 0.0137933i
$353$	30.2075i	1.60779i	0.594775	+	0.803893i	$0.297241\pi$
$353$	30.2075i	1.60779i	−0.594775	+	0.803893i	$0.702759\pi$
$354$	−28.1530	−1.49631
$355$	0	0
$356$	6.00000	0.317999
$357$	− 16.5176i	− 0.874203i
$358$	8.73501i	0.461660i
$359$	7.39973	0.390543	0.195271	−	0.980749i	$-0.437441\pi$
$359$	7.39973	0.390543	0.195271	+	0.980749i	$0.437441\pi$
$360$	0	0
$361$	−10.3609	−0.545310
$362$	10.6391i	0.559179i
$363$	32.1347i	1.68664i
$364$	−7.75694	−0.406574
$365$	0	0
$366$	1.11784	0.0584303
$367$	− 13.9198i	− 0.726609i	−0.931671	−	0.363304i	$-0.881649\pi$
$367$	− 13.9198i	− 0.726609i	0.931671	−	0.363304i	$-0.118351\pi$
$368$	8.51757i	0.444009i
$369$	50.1761	2.61206
$370$	0	0
$371$	−17.0194	−0.883604
$372$	11.6354i	0.603268i
$373$	− 20.2357i	− 1.04776i	−0.851791	−	0.523882i	$-0.824483\pi$
$373$	− 20.2357i	− 1.04776i	0.851791	−	0.523882i	$-0.175517\pi$
$374$	1.10211	0.0569889
$375$	0	0
$376$	5.57834	0.287681
$377$	21.2782i	1.09588i
$378$	10.2357i	0.526466i
$379$	27.9488	1.43563	0.717816	−	0.696233i	$-0.245142\pi$
$379$	27.9488	1.43563	0.717816	+	0.696233i	$0.245142\pi$
$380$	0	0
$381$	21.9173	1.12286
$382$	22.1116i	1.13133i
$383$	− 11.7569i	− 0.600752i	−0.953821	−	0.300376i	$-0.902888\pi$
$383$	− 11.7569i	− 0.600752i	0.953821	−	0.300376i	$-0.0971121\pi$
$384$	−2.93923	−0.149992
$385$	0	0
$386$	7.03514	0.358079
$387$	− 31.6900i	− 1.61089i
$388$	12.8979i	0.654791i
$389$	9.58085	0.485768	0.242884	−	0.970055i	$-0.421907\pi$
$389$	9.58085	0.485768	0.242884	+	0.970055i	$0.421907\pi$
$390$	0	0
$391$	−36.2745	−1.83448
$392$	− 5.25879i	− 0.265609i
$393$	− 7.11784i	− 0.359047i
$394$	−25.7569	−1.29762
$395$	0	0
$396$	−1.45932	−0.0733335
$397$	− 32.4787i	− 1.63006i	−0.579418	−	0.815031i	$-0.696720\pi$
$397$	− 32.4787i	− 1.63006i	0.579418	−	0.815031i	$-0.303280\pi$
$398$	12.7350i	0.638348i
$399$	11.3997	0.570700
$400$	0	0
$401$	7.27820	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$401$	7.27820	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$402$	16.3960i	0.817760i
$403$	23.2708i	1.15920i
$404$	−10.3960	−0.517222
$405$	0	0
$406$	4.77636	0.237047
$407$	0.258786i	0.0128275i
$408$	− 12.5176i	− 0.619712i
$409$	32.9963	1.63156	0.815781	−	0.578361i	$-0.196307\pi$
$409$	32.9963	1.63156	0.815781	+	0.578361i	$0.196307\pi$
$410$	0	0
$411$	46.4275	2.29010
$412$	− 13.8785i	− 0.683743i
$413$	12.6391i	0.621930i
$414$	48.0315	2.36062
$415$	0	0
$416$	−5.87847	−0.288216
$417$	− 33.5527i	− 1.64308i
$418$	0.760632i	0.0372037i
$419$	−25.7131	−1.25617	−0.628083	−	0.778146i	$-0.716160\pi$
$419$	−25.7131	−1.25617	−0.628083	+	0.778146i	$0.716160\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 6.65483i	− 0.323952i
$423$	− 31.4568i	− 1.52948i
$424$	−12.8979	−0.626377
$425$	0	0
$426$	33.0351	1.60056
$427$	− 0.501846i	− 0.0242860i
$428$	4.21744i	0.203858i
$429$	−4.47135	−0.215879
$430$	0	0
$431$	17.9198	0.863167	0.431584	−	0.902073i	$-0.357955\pi$
$431$	17.9198	0.863167	0.431584	+	0.902073i	$0.357955\pi$
$432$	7.75694i	0.373206i
$433$	− 4.96486i	− 0.238596i	−0.992859	−	0.119298i	$-0.961936\pi$
$433$	− 4.96486i	− 0.238596i	0.992859	−	0.119298i	$-0.0380644\pi$
$434$	5.22364	0.250743
$435$	0	0
$436$	16.1373	0.772834
$437$	− 25.0351i	− 1.19759i
$438$	− 7.39973i	− 0.353573i
$439$	13.8371	0.660410	0.330205	−	0.943909i	$-0.392882\pi$
$439$	13.8371	0.660410	0.330205	+	0.943909i	$0.392882\pi$
$440$	0	0
$441$	−29.6548	−1.41213
$442$	− 25.0351i	− 1.19080i
$443$	− 14.3390i	− 0.681265i	−0.940197	−	0.340632i	$-0.889359\pi$
$443$	− 14.3390i	− 0.681265i	0.940197	−	0.340632i	$-0.110641\pi$
$444$	2.93923	0.139490
$445$	0	0
$446$	−16.5589	−0.784088
$447$	54.0703i	2.55744i
$448$	1.31955i	0.0623429i
$449$	−12.4787	−0.588908	−0.294454	−	0.955666i	$-0.595138\pi$
$449$	−12.4787	−0.588908	−0.294454	+	0.955666i	$0.595138\pi$
$450$	0	0
$451$	2.30265	0.108427
$452$	1.01942i	0.0479494i
$453$	− 17.2782i	− 0.811801i
$454$	4.01573	0.188467
$455$	0	0
$456$	8.63910	0.404563
$457$	21.6585i	1.01314i	0.862198	+	0.506571i	$0.169087\pi$
$457$	21.6585i	1.01314i	−0.862198	+	0.506571i	$0.830913\pi$
$458$	− 25.5139i	− 1.19219i
$459$	−33.0351	−1.54195
$460$	0	0
$461$	−29.4155	−1.37001	−0.685007	−	0.728536i	$-0.740201\pi$
$461$	−29.4155	−1.37001	−0.685007	+	0.728536i	$0.740201\pi$
$462$	1.00369i	0.0466960i
$463$	9.03514i	0.419899i	0.977712	+	0.209949i	$0.0673299\pi$
$463$	9.03514i	0.419899i	−0.977712	+	0.209949i	$0.932670\pi$
$464$	3.61968	0.168040
$465$	0	0
$466$	7.96116	0.368794
$467$	7.05825i	0.326617i	0.986575	+	0.163308i	$0.0522166\pi$
$467$	7.05825i	0.326617i	−0.986575	+	0.163308i	$0.947783\pi$
$468$	33.1493i	1.53233i
$469$	7.36090	0.339895
$470$	0	0
$471$	10.6391	0.490224
$472$	9.57834i	0.440879i
$473$	− 1.45429i	− 0.0668685i
$474$	22.6317	1.03951
$475$	0	0
$476$	−5.61968	−0.257578
$477$	72.7325i	3.33019i
$478$	− 11.7983i	− 0.539641i
$479$	29.3353	1.34036	0.670181	−	0.742197i	$-0.266216\pi$
$479$	29.3353	1.34036	0.670181	+	0.742197i	$0.266216\pi$
$480$	0	0
$481$	5.87847	0.268035
$482$	24.9963i	1.13855i
$483$	− 33.0351i	− 1.50315i
$484$	10.9330	0.496956
$485$	0	0
$486$	−5.98176	−0.271338
$487$	− 37.9099i	− 1.71786i	−0.512091	−	0.858931i	$-0.671129\pi$
$487$	− 37.9099i	− 1.71786i	0.512091	−	0.858931i	$-0.328871\pi$
$488$	− 0.380316i	− 0.0172161i
$489$	64.8235	2.93142
$490$	0	0
$491$	−14.5564	−0.656921	−0.328461	−	0.944518i	$-0.606530\pi$
$491$	−14.5564	−0.656921	−0.328461	+	0.944518i	$0.606530\pi$
$492$	− 26.1530i	− 1.17907i
$493$	15.4155i	0.694277i
$494$	17.2782	0.777383
$495$	0	0
$496$	3.95865	0.177749
$497$	− 14.8309i	− 0.665258i
$498$	18.1604i	0.813785i
$499$	−23.9744	−1.07324	−0.536620	−	0.843824i	$-0.680299\pi$
$499$	−23.9744	−1.07324	−0.536620	+	0.843824i	$0.680299\pi$
$500$	0	0
$501$	−1.76432	−0.0788242
$502$	− 8.81770i	− 0.393553i
$503$	43.7569i	1.95103i	0.219943	+	0.975513i	$0.429413\pi$
$503$	43.7569i	1.95103i	−0.219943	+	0.975513i	$0.570587\pi$
$504$	7.44108	0.331452
$505$	0	0
$506$	2.20423	0.0979898
$507$	63.3593i	2.81389i
$508$	− 7.45681i	− 0.330842i
$509$	−11.7958	−0.522839	−0.261419	−	0.965225i	$-0.584191\pi$
$509$	−11.7958	−0.522839	−0.261419	+	0.965225i	$0.584191\pi$
$510$	0	0
$511$	−3.32206	−0.146959
$512$	1.00000i	0.0441942i
$513$	− 22.7995i	− 1.00662i
$514$	−11.8396	−0.522224
$515$	0	0
$516$	−16.5176	−0.727146
$517$	− 1.44359i	− 0.0634892i
$518$	− 1.31955i	− 0.0579777i
$519$	66.7020	2.92789
$520$	0	0
$521$	14.9806	0.656311	0.328156	−	0.944624i	$-0.393573\pi$
$521$	14.9806	0.656311	0.328156	+	0.944624i	$0.393573\pi$
$522$	− 20.4118i	− 0.893399i
$523$	27.5139i	1.20310i	0.798836	+	0.601549i	$0.205450\pi$
$523$	27.5139i	1.20310i	−0.798836	+	0.601549i	$0.794550\pi$
$524$	−2.42166	−0.105791
$525$	0	0
$526$	−9.15919	−0.399359
$527$	16.8591i	0.734392i
$528$	0.760632i	0.0331023i
$529$	−49.5490	−2.15431
$530$	0	0
$531$	54.0132	2.34397
$532$	− 3.87847i	− 0.168153i
$533$	− 52.3060i	− 2.26562i
$534$	−17.6354	−0.763159
$535$	0	0
$536$	5.57834	0.240947
$537$	− 25.6742i	− 1.10793i
$538$	− 18.3960i	− 0.793110i
$539$	−1.36090	−0.0586180
$540$	0	0
$541$	−11.0351	−0.474438	−0.237219	−	0.971456i	$-0.576236\pi$
$541$	−11.0351	−0.474438	−0.237219	+	0.971456i	$0.576236\pi$
$542$	− 27.2394i	− 1.17003i
$543$	− 31.2708i	− 1.34196i
$544$	−4.25879	−0.182594
$545$	0	0
$546$	22.7995	0.975727
$547$	− 16.1761i	− 0.691640i	−0.938301	−	0.345820i	$-0.887601\pi$
$547$	− 16.1761i	− 0.691640i	0.938301	−	0.345820i	$-0.112399\pi$
$548$	− 15.7958i	− 0.674762i
$549$	−2.14464	−0.0915310
$550$	0	0
$551$	−10.6391	−0.453241
$552$	− 25.0351i	− 1.06557i
$553$	− 10.1604i	− 0.432063i
$554$	21.8785	0.929527
$555$	0	0
$556$	−11.4155	−0.484123
$557$	11.1567i	0.472723i	0.971665	+	0.236362i	$0.0759550\pi$
$557$	11.1567i	0.472723i	−0.971665	+	0.236362i	$0.924045\pi$
$558$	− 22.3232i	− 0.945018i
$559$	−33.0351	−1.39724
$560$	0	0
$561$	−3.23937	−0.136766
$562$	− 20.9963i	− 0.885676i
$563$	41.7288i	1.75866i	0.476213	+	0.879330i	$0.342009\pi$
$563$	41.7288i	1.75866i	−0.476213	+	0.879330i	$0.657991\pi$
$564$	−16.3960	−0.690398
$565$	0	0
$566$	18.9136	0.794998
$567$	− 7.76181i	− 0.325965i
$568$	− 11.2394i	− 0.471593i
$569$	24.0703	1.00908	0.504539	−	0.863389i	$-0.331662\pi$
$569$	24.0703	1.00908	0.504539	+	0.863389i	$0.331662\pi$
$570$	0	0
$571$	37.6511	1.57565	0.787825	−	0.615899i	$-0.211207\pi$
$571$	37.6511	1.57565	0.787825	+	0.615899i	$0.211207\pi$
$572$	1.52126i	0.0636072i
$573$	− 64.9913i	− 2.71505i
$574$	−11.7412	−0.490069
$575$	0	0
$576$	5.63910	0.234963
$577$	23.4312i	0.975453i	0.872996	+	0.487726i	$0.162174\pi$
$577$	23.4312i	0.975453i	−0.872996	+	0.487726i	$0.837826\pi$
$578$	− 1.13726i	− 0.0473036i
$579$	−20.6779	−0.859346
$580$	0	0
$581$	8.15298	0.338243
$582$	− 37.9099i	− 1.57142i
$583$	3.33779i	0.138237i
$584$	−2.51757	−0.104178
$585$	0	0
$586$	−12.8979	−0.532807
$587$	6.30265i	0.260138i	0.991505	+	0.130069i	$0.0415199\pi$
$587$	6.30265i	0.260138i	−0.991505	+	0.130069i	$0.958480\pi$
$588$	15.4568i	0.637428i
$589$	−11.6354	−0.479429
$590$	0	0
$591$	75.7057	3.11412
$592$	− 1.00000i	− 0.0410997i
$593$	34.0315i	1.39750i	0.715364	+	0.698752i	$0.246261\pi$
$593$	34.0315i	1.39750i	−0.715364	+	0.698752i	$0.753739\pi$
$594$	2.00738	0.0823640
$595$	0	0
$596$	18.3960	0.753531
$597$	− 37.4312i	− 1.53196i
$598$	− 50.0703i	− 2.04753i
$599$	−34.8359	−1.42336	−0.711679	−	0.702505i	$-0.752065\pi$
$599$	−34.8359	−1.42336	−0.711679	+	0.702505i	$0.752065\pi$
$600$	0	0
$601$	15.2236	0.620985	0.310493	−	0.950576i	$-0.399506\pi$
$601$	15.2236	0.620985	0.310493	+	0.950576i	$0.399506\pi$
$602$	7.41546i	0.302232i
$603$	− 31.4568i	− 1.28102i
$604$	−5.87847	−0.239192
$605$	0	0
$606$	30.5564	1.24127
$607$	− 30.8309i	− 1.25139i	−0.780068	−	0.625694i	$-0.784816\pi$
$607$	− 30.8309i	− 1.25139i	0.780068	−	0.625694i	$-0.215184\pi$
$608$	− 2.93923i	− 0.119202i
$609$	−14.0388	−0.568882
$610$	0	0
$611$	−32.7921	−1.32663
$612$	24.0157i	0.970778i
$613$	11.2112i	0.452817i	0.974032	+	0.226409i	$0.0726985\pi$
$613$	11.2112i	0.452817i	−0.974032	+	0.226409i	$0.927302\pi$
$614$	3.78256	0.152652
$615$	0	0
$616$	0.341481	0.0137587
$617$	26.5176i	1.06756i	0.845624	+	0.533779i	$0.179228\pi$
$617$	26.5176i	1.06756i	−0.845624	+	0.533779i	$0.820772\pi$
$618$	40.7921i	1.64090i
$619$	3.41546	0.137279	0.0686394	−	0.997642i	$-0.478134\pi$
$619$	3.41546	0.137279	0.0686394	+	0.997642i	$0.478134\pi$
$620$	0	0
$621$	−66.0703	−2.65131
$622$	− 15.2807i	− 0.612701i
$623$	7.91730i	0.317200i
$624$	17.2782	0.691682
$625$	0	0
$626$	19.0351	0.760797
$627$	− 2.23568i	− 0.0892843i
$628$	− 3.61968i	− 0.144441i
$629$	4.25879	0.169809
$630$	0	0
$631$	3.11533	0.124019	0.0620096	−	0.998076i	$-0.480249\pi$
$631$	3.11533	0.124019	0.0620096	+	0.998076i	$0.480249\pi$
$632$	− 7.69987i	− 0.306284i
$633$	19.5601i	0.777444i
$634$	24.1373	0.958613
$635$	0	0
$636$	37.9099	1.50323
$637$	30.9136i	1.22484i
$638$	− 0.936722i	− 0.0370852i
$639$	−63.3799	−2.50727
$640$	0	0
$641$	−20.5018	−0.809774	−0.404887	−	0.914367i	$-0.632689\pi$
$641$	−20.5018	−0.809774	−0.404887	+	0.914367i	$0.632689\pi$
$642$	− 12.3960i	− 0.489233i
$643$	− 5.70238i	− 0.224880i	−0.993659	−	0.112440i	$-0.964133\pi$
$643$	− 5.70238i	− 0.224880i	0.993659	−	0.112440i	$-0.0358666\pi$
$644$	−11.2394	−0.442893
$645$	0	0
$646$	12.5176	0.492497
$647$	− 41.0351i	− 1.61326i	−0.591058	−	0.806629i	$-0.701290\pi$
$647$	− 41.0351i	− 1.61326i	0.591058	−	0.806629i	$-0.298710\pi$
$648$	− 5.88216i	− 0.231073i
$649$	2.47874	0.0972989
$650$	0	0
$651$	−15.3535	−0.601752
$652$	− 22.0546i	− 0.863723i
$653$	− 14.9575i	− 0.585331i	−0.956215	−	0.292666i	$-0.905458\pi$
$653$	− 14.9575i	− 0.585331i	0.956215	−	0.292666i	$-0.0945423\pi$
$654$	−47.4312	−1.85471
$655$	0	0
$656$	−8.89789	−0.347404
$657$	14.1968i	0.553872i
$658$	7.36090i	0.286958i
$659$	15.7569	0.613803	0.306902	−	0.951741i	$-0.400708\pi$
$659$	15.7569	0.613803	0.306902	+	0.951741i	$0.400708\pi$
$660$	0	0
$661$	11.6197	0.451953	0.225977	−	0.974133i	$-0.427443\pi$
$661$	11.6197	0.451953	0.225977	+	0.974133i	$0.427443\pi$
$662$	0.817705i	0.0317810i
$663$	73.5842i	2.85777i
$664$	6.17860	0.239776
$665$	0	0
$666$	−5.63910	−0.218511
$667$	30.8309i	1.19378i
$668$	0.600267i	0.0232250i
$669$	48.6706	1.88171
$670$	0	0
$671$	−0.0984203	−0.00379947
$672$	− 3.87847i	− 0.149615i
$673$	− 12.7218i	− 0.490389i	−0.969474	−	0.245195i	$-0.921148\pi$
$673$	− 12.7218i	− 0.490389i	0.969474	−	0.245195i	$-0.0788519\pi$
$674$	19.7958	0.762505
$675$	0	0
$676$	21.5564	0.829093
$677$	− 21.5139i	− 0.826846i	−0.910539	−	0.413423i	$-0.864333\pi$
$677$	− 21.5139i	− 0.826846i	0.910539	−	0.413423i	$-0.135667\pi$
$678$	− 2.99631i	− 0.115073i
$679$	−17.0194	−0.653145
$680$	0	0
$681$	−11.8032	−0.452298
$682$	− 1.02444i	− 0.0392279i
$683$	− 47.6900i	− 1.82481i	−0.409293	−	0.912403i	$-0.634225\pi$
$683$	− 47.6900i	− 1.82481i	0.409293	−	0.912403i	$-0.365775\pi$
$684$	−16.5746	−0.633747
$685$	0	0
$686$	16.1761	0.617606
$687$	74.9913i	2.86110i
$688$	5.61968i	0.214248i
$689$	75.8198	2.88851
$690$	0	0
$691$	30.5721	1.16302	0.581509	−	0.813540i	$-0.302462\pi$
$691$	30.5721	1.16302	0.581509	+	0.813540i	$0.302462\pi$
$692$	− 22.6937i	− 0.862684i
$693$	− 1.92565i	− 0.0731492i
$694$	−15.7569	−0.598125
$695$	0	0
$696$	−10.6391	−0.403274
$697$	− 37.8942i	− 1.43534i
$698$	0.160365i	0.00606992i
$699$	−23.3997	−0.885059
$700$	0	0
$701$	32.0703	1.21128	0.605639	−	0.795740i	$-0.292918\pi$
$701$	32.0703	1.21128	0.605639	+	0.795740i	$0.292918\pi$
$702$	− 45.5989i	− 1.72102i
$703$	2.93923i	0.110855i
$704$	0.258786	0.00975335
$705$	0	0
$706$	−30.2075	−1.13688
$707$	− 13.7181i	− 0.515922i
$708$	− 28.1530i	− 1.05805i
$709$	−10.6160	−0.398692	−0.199346	−	0.979929i	$-0.563882\pi$
$709$	−10.6160	−0.398692	−0.199346	+	0.979929i	$0.563882\pi$
$710$	0	0
$711$	−43.4203	−1.62839
$712$	6.00000i	0.224860i
$713$	33.7181i	1.26275i
$714$	16.5176	0.618155
$715$	0	0
$716$	−8.73501	−0.326443
$717$	34.6779i	1.29507i
$718$	7.39973i	0.276156i
$719$	32.4663	1.21079	0.605395	−	0.795925i	$-0.293015\pi$
$719$	32.4663	1.21079	0.605395	+	0.795925i	$0.293015\pi$
$720$	0	0
$721$	18.3133	0.682025
$722$	− 10.3609i	− 0.385593i
$723$	− 73.4700i	− 2.73238i
$724$	−10.6391	−0.395399
$725$	0	0
$726$	−32.1347	−1.19263
$727$	31.5139i	1.16879i	0.811471	+	0.584393i	$0.198667\pi$
$727$	31.5139i	1.16879i	−0.811471	+	0.584393i	$0.801333\pi$
$728$	− 7.75694i	− 0.287491i
$729$	35.2283	1.30475
$730$	0	0
$731$	−23.9330	−0.885195
$732$	1.11784i	0.0413165i
$733$	16.8202i	0.621269i	0.950529	+	0.310634i	$0.100541\pi$
$733$	16.8202i	0.621269i	−0.950529	+	0.310634i	$0.899459\pi$
$734$	13.9198	0.513790
$735$	0	0
$736$	−8.51757	−0.313962
$737$	− 1.44359i	− 0.0531755i
$738$	50.1761i	1.84701i
$739$	−9.69735	−0.356723	−0.178361	−	0.983965i	$-0.557080\pi$
$739$	−9.69735	−0.356723	−0.178361	+	0.983965i	$0.557080\pi$
$740$	0	0
$741$	−50.7847	−1.86562
$742$	− 17.0194i	− 0.624802i
$743$	39.1153i	1.43500i	0.696557	+	0.717501i	$0.254714\pi$
$743$	39.1153i	1.43500i	−0.696557	+	0.717501i	$0.745286\pi$
$744$	−11.6354	−0.426575
$745$	0	0
$746$	20.2357	0.740881
$747$	− 34.8418i	− 1.27479i
$748$	1.10211i	0.0402972i
$749$	−5.56512	−0.203345
$750$	0	0
$751$	−26.1530	−0.954336	−0.477168	−	0.878812i	$-0.658337\pi$
$751$	−26.1530	−0.954336	−0.477168	+	0.878812i	$0.658337\pi$
$752$	5.57834i	0.203421i
$753$	25.9173i	0.944479i
$754$	−21.2782	−0.774906
$755$	0	0
$756$	−10.2357	−0.372268
$757$	− 33.5915i	− 1.22091i	−0.792053	−	0.610453i	$-0.790987\pi$
$757$	− 33.5915i	− 1.22091i	0.792053	−	0.610453i	$-0.209013\pi$
$758$	27.9488i	1.01514i
$759$	−6.47874	−0.235163
$760$	0	0
$761$	−51.3766	−1.86240	−0.931201	−	0.364507i	$-0.881237\pi$
$761$	−51.3766	−1.86240	−0.931201	+	0.364507i	$0.881237\pi$
$762$	21.9173i	0.793980i
$763$	21.2939i	0.770892i
$764$	−22.1116	−0.799971
$765$	0	0
$766$	11.7569	0.424795
$767$	− 56.3060i	− 2.03309i
$768$	− 2.93923i	− 0.106061i
$769$	−27.5527	−0.993576	−0.496788	−	0.867872i	$-0.665487\pi$
$769$	−27.5527	−0.993576	−0.496788	+	0.867872i	$0.665487\pi$
$770$	0	0
$771$	34.7995	1.25327
$772$	7.03514i	0.253200i
$773$	− 13.4155i	− 0.482521i	−0.970460	−	0.241260i	$-0.922439\pi$
$773$	− 13.4155i	− 0.482521i	0.970460	−	0.241260i	$-0.0775607\pi$
$774$	31.6900	1.13907
$775$	0	0
$776$	−12.8979	−0.463007
$777$	3.87847i	0.139139i
$778$	9.58085i	0.343490i
$779$	26.1530	0.937028
$780$	0	0
$781$	−2.90859	−0.104077
$782$	− 36.2745i	− 1.29717i
$783$	28.0777i	1.00341i
$784$	5.25879	0.187814
$785$	0	0
$786$	7.11784	0.253885
$787$	− 25.4956i	− 0.908821i	−0.890792	−	0.454411i	$-0.849850\pi$
$787$	− 25.4956i	− 0.908821i	0.890792	−	0.454411i	$-0.150150\pi$
$788$	− 25.7569i	− 0.917553i
$789$	26.9210	0.958413
$790$	0	0
$791$	−1.34517	−0.0478289
$792$	− 1.45932i	− 0.0518546i
$793$	2.23568i	0.0793912i
$794$	32.4787	1.15263
$795$	0	0
$796$	−12.7350	−0.451380
$797$	48.4663i	1.71677i	0.513010	+	0.858383i	$0.328530\pi$
$797$	48.4663i	1.71677i	−0.513010	+	0.858383i	$0.671470\pi$
$798$	11.3997i	0.403546i
$799$	−23.7569	−0.840460
$800$	0	0
$801$	33.8346	1.19549
$802$	7.27820i	0.257002i
$803$	0.651511i	0.0229913i
$804$	−16.3960	−0.578244
$805$	0	0
$806$	−23.2708	−0.819680
$807$	54.0703i	1.90336i
$808$	− 10.3960i	− 0.365731i
$809$	12.9649	0.455820	0.227910	−	0.973682i	$-0.426811\pi$
$809$	12.9649	0.455820	0.227910	+	0.973682i	$0.426811\pi$
$810$	0	0
$811$	0.0776702	0.00272737	0.00136368	−	0.999999i	$-0.499566\pi$
$811$	0.0776702	0.00272737	0.00136368	+	0.999999i	$0.499566\pi$
$812$	4.77636i	0.167617i
$813$	80.0629i	2.80793i
$814$	−0.258786	−0.00907043
$815$	0	0
$816$	12.5176	0.438203
$817$	− 16.5176i	− 0.577877i
$818$	32.9963i	1.15369i
$819$	−43.7422	−1.52848
$820$	0	0
$821$	36.3448	1.26844	0.634221	−	0.773152i	$-0.281321\pi$
$821$	36.3448	1.26844	0.634221	+	0.773152i	$0.281321\pi$
$822$	46.4275i	1.61934i
$823$	23.0996i	0.805201i	0.915376	+	0.402601i	$0.131894\pi$
$823$	23.0996i	0.805201i	−0.915376	+	0.402601i	$0.868106\pi$
$824$	13.8785	0.483479
$825$	0	0
$826$	−12.6391	−0.439771
$827$	15.2551i	0.530472i	0.964184	+	0.265236i	$0.0854498\pi$
$827$	15.2551i	0.530472i	−0.964184	+	0.265236i	$0.914550\pi$
$828$	48.0315i	1.66921i
$829$	−28.1373	−0.977247	−0.488624	−	0.872495i	$-0.662501\pi$
$829$	−28.1373	−0.977247	−0.488624	+	0.872495i	$0.662501\pi$
$830$	0	0
$831$	−64.3060	−2.23075
$832$	− 5.87847i	− 0.203799i
$833$	22.3960i	0.775977i
$834$	33.5527	1.16184
$835$	0	0
$836$	−0.760632	−0.0263070
$837$	30.7070i	1.06139i
$838$	− 25.7131i	− 0.888244i
$839$	−4.24306	−0.146487	−0.0732434	−	0.997314i	$-0.523335\pi$
$839$	−4.24306	−0.146487	−0.0732434	+	0.997314i	$0.523335\pi$
$840$	0	0
$841$	−15.8979	−0.548203
$842$	22.0000i	0.758170i
$843$	61.7131i	2.12551i
$844$	6.65483	0.229069
$845$	0	0
$846$	31.4568	1.08151
$847$	14.4267i	0.495707i
$848$	− 12.8979i	− 0.442915i
$849$	−55.5915	−1.90790
$850$	0	0
$851$	8.51757	0.291979
$852$	33.0351i	1.13177i
$853$	− 36.4787i	− 1.24901i	−0.781022	−	0.624504i	$-0.785301\pi$
$853$	− 36.4787i	− 1.24901i	0.781022	−	0.624504i	$-0.214699\pi$
$854$	0.501846	0.0171728
$855$	0	0
$856$	−4.21744	−0.144149
$857$	23.7288i	0.810561i	0.914192	+	0.405280i	$0.132826\pi$
$857$	23.7288i	0.810561i	−0.914192	+	0.405280i	$0.867174\pi$
$858$	− 4.47135i	− 0.152649i
$859$	−2.77887	−0.0948138	−0.0474069	−	0.998876i	$-0.515096\pi$
$859$	−2.77887	−0.0948138	−0.0474069	+	0.998876i	$0.515096\pi$
$860$	0	0
$861$	34.5102	1.17610
$862$	17.9198i	0.610351i
$863$	− 10.5151i	− 0.357937i	−0.983855	−	0.178968i	$-0.942724\pi$
$863$	− 10.5151i	− 0.357937i	0.983855	−	0.178968i	$-0.0572760\pi$
$864$	−7.75694	−0.263896
$865$	0	0
$866$	4.96486	0.168713
$867$	3.34266i	0.113523i
$868$	5.22364i	0.177302i
$869$	−1.99262	−0.0675948
$870$	0	0
$871$	−32.7921	−1.11112
$872$	16.1373i	0.546476i
$873$	72.7325i	2.46162i
$874$	25.0351	0.846826
$875$	0	0
$876$	7.39973	0.250014
$877$	27.6197i	0.932650i	0.884613	+	0.466325i	$0.154422\pi$
$877$	27.6197i	0.932650i	−0.884613	+	0.466325i	$0.845578\pi$
$878$	13.8371i	0.466980i
$879$	37.9099	1.27867
$880$	0	0
$881$	10.2200	0.344319	0.172159	−	0.985069i	$-0.444926\pi$
$881$	10.2200	0.344319	0.172159	+	0.985069i	$0.444926\pi$
$882$	− 29.6548i	− 0.998530i
$883$	− 53.5684i	− 1.80272i	−0.433069	−	0.901361i	$-0.642569\pi$
$883$	− 53.5684i	− 1.80272i	0.433069	−	0.901361i	$-0.357431\pi$
$884$	25.0351	0.842023
$885$	0	0
$886$	14.3390	0.481727
$887$	36.5904i	1.22858i	0.789079	+	0.614292i	$0.210558\pi$
$887$	36.5904i	1.22858i	−0.789079	+	0.614292i	$0.789442\pi$
$888$	2.93923i	0.0986343i
$889$	9.83963	0.330011
$890$	0	0
$891$	−1.52222	−0.0509963
$892$	− 16.5589i	− 0.554434i
$893$	− 16.3960i	− 0.548673i
$894$	−54.0703	−1.80838
$895$	0	0
$896$	−1.31955	−0.0440831
$897$	147.168i	4.91381i
$898$	− 12.4787i	− 0.416421i
$899$	14.3291	0.477901
$900$	0	0
$901$	54.9293	1.82996
$902$	2.30265i	0.0766697i
$903$	− 21.7958i	− 0.725318i
$904$	−1.01942	−0.0339053
$905$	0	0
$906$	17.2782	0.574030
$907$	7.64279i	0.253775i	0.991917	+	0.126887i	$0.0404987\pi$
$907$	7.64279i	0.253775i	−0.991917	+	0.126887i	$0.959501\pi$
$908$	4.01573i	0.133267i
$909$	−58.6243	−1.94445
$910$	0	0
$911$	22.0132	0.729330	0.364665	−	0.931139i	$-0.381183\pi$
$911$	22.0132	0.729330	0.364665	+	0.931139i	$0.381183\pi$
$912$	8.63910i	0.286069i
$913$	− 1.59893i	− 0.0529170i
$914$	−21.6585	−0.716400
$915$	0	0
$916$	25.5139	0.843002
$917$	− 3.19551i	− 0.105525i
$918$	− 33.0351i	− 1.09032i
$919$	−23.6999	−0.781786	−0.390893	−	0.920436i	$-0.627834\pi$
$919$	−23.6999	−0.781786	−0.390893	+	0.920436i	$0.627834\pi$
$920$	0	0
$921$	−11.1178	−0.366345
$922$	− 29.4155i	− 0.968747i
$923$	66.0703i	2.17473i
$924$	−1.00369	−0.0330191
$925$	0	0
$926$	−9.03514	−0.296913
$927$	− 78.2621i	− 2.57046i
$928$	3.61968i	0.118822i
$929$	−8.82022	−0.289382	−0.144691	−	0.989477i	$-0.546219\pi$
$929$	−8.82022	−0.289382	−0.144691	+	0.989477i	$0.546219\pi$
$930$	0	0
$931$	−15.4568	−0.506576
$932$	7.96116i	0.260777i
$933$	44.9136i	1.47041i
$934$	−7.05825	−0.230953
$935$	0	0
$936$	−33.1493	−1.08352
$937$	40.0703i	1.30904i	0.756045	+	0.654520i	$0.227129\pi$
$937$	40.0703i	1.30904i	−0.756045	+	0.654520i	$0.772871\pi$
$938$	7.36090i	0.240342i
$939$	−55.9488	−1.82582
$940$	0	0
$941$	−31.4312	−1.02463	−0.512314	−	0.858798i	$-0.671211\pi$
$941$	−31.4312	−1.02463	−0.512314	+	0.858798i	$0.671211\pi$
$942$	10.6391i	0.346641i
$943$	− 75.7884i	− 2.46801i
$944$	−9.57834	−0.311748
$945$	0	0
$946$	1.45429	0.0472832
$947$	− 53.5370i	− 1.73972i	−0.493300	−	0.869859i	$-0.664210\pi$
$947$	− 53.5370i	− 1.73972i	0.493300	−	0.869859i	$-0.335790\pi$
$948$	22.6317i	0.735044i
$949$	14.7995	0.480411
$950$	0	0
$951$	−70.9451	−2.30055
$952$	− 5.61968i	− 0.182135i
$953$	− 5.23937i	− 0.169720i	−0.996393	−	0.0848599i	$-0.972956\pi$
$953$	− 5.23937i	− 0.169720i	0.996393	−	0.0848599i	$-0.0270443\pi$
$954$	−72.7325	−2.35480
$955$	0	0
$956$	11.7983	0.381584
$957$	2.75325i	0.0889998i
$958$	29.3353i	0.947780i
$959$	20.8433	0.673066
$960$	0	0
$961$	−15.3291	−0.494486
$962$	5.87847i	0.189529i
$963$	23.7826i	0.766382i
$964$	−24.9963	−0.805077
$965$	0	0
$966$	33.0351	1.06289
$967$	− 46.5929i	− 1.49833i	−0.662386	−	0.749163i	$-0.730456\pi$
$967$	− 46.5929i	− 1.49833i	0.662386	−	0.749163i	$-0.269544\pi$
$968$	10.9330i	0.351401i
$969$	−36.7921	−1.18193
$970$	0	0
$971$	38.1373	1.22388	0.611941	−	0.790903i	$-0.290389\pi$
$971$	38.1373	1.22388	0.611941	+	0.790903i	$0.290389\pi$
$972$	− 5.98176i	− 0.191865i
$973$	− 15.0633i	− 0.482907i
$974$	37.9099	1.21471
$975$	0	0
$976$	0.380316	0.0121736
$977$	40.8979i	1.30844i	0.756305	+	0.654220i	$0.227003\pi$
$977$	40.8979i	1.30844i	−0.756305	+	0.654220i	$0.772997\pi$
$978$	64.8235i	2.07283i
$979$	1.55271	0.0496250
$980$	0	0
$981$	90.9996	2.90539
$982$	− 14.5564i	− 0.464514i
$983$	− 50.4638i	− 1.60955i	−0.593583	−	0.804773i	$-0.702287\pi$
$983$	− 50.4638i	− 1.60955i	0.593583	−	0.804773i	$-0.297713\pi$
$984$	26.1530	0.833727
$985$	0	0
$986$	−15.4155	−0.490928
$987$	− 21.6354i	− 0.688663i
$988$	17.2782i	0.549693i
$989$	−47.8661	−1.52205
$990$	0	0
$991$	−15.9587	−0.506943	−0.253472	−	0.967343i	$-0.581572\pi$
$991$	−15.9587	−0.506943	−0.253472	+	0.967343i	$0.581572\pi$
$992$	3.95865i	0.125687i
$993$	− 2.40343i	− 0.0762704i
$994$	14.8309	0.470408
$995$	0	0
$996$	−18.1604	−0.575433
$997$	− 9.23937i	− 0.292614i	−0.989239	−	0.146307i	$-0.953261\pi$
$997$	− 9.23937i	− 0.292614i	0.989239	−	0.146307i	$-0.0467387\pi$
$998$	− 23.9744i	− 0.758896i
$999$	7.75694	0.245419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.o.149.4		6
5.2	odd	4		1850.2.a.z.1.1		3
5.3	odd	4		370.2.a.g.1.3	✓	3
5.4	even	2	inner	1850.2.b.o.149.3		6
15.8	even	4		3330.2.a.bg.1.2		3
20.3	even	4		2960.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.3	✓	3	5.3	odd	4
1850.2.a.z.1.1		3	5.2	odd	4
1850.2.b.o.149.3		6	5.4	even	2	inner
1850.2.b.o.149.4		6	1.1	even	1	trivial
2960.2.a.u.1.1		3	20.3	even	4
3330.2.a.bg.1.2		3	15.8	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.93923i q^{3} -1.00000 q^{4} +2.93923 q^{6} -1.31955i q^{7} -1.00000i q^{8} -5.63910 q^{9} -0.258786 q^{11} +2.93923i q^{12} +5.87847i q^{13} +1.31955 q^{14} +1.00000 q^{16} +4.25879i q^{17} -5.63910i q^{18} -2.93923 q^{19} -3.87847 q^{21} -0.258786i q^{22} +8.51757i q^{23} -2.93923 q^{24} -5.87847 q^{26} +7.75694i q^{27} +1.31955i q^{28} +3.61968 q^{29} +3.95865 q^{31} +1.00000i q^{32} +0.760632i q^{33} -4.25879 q^{34} +5.63910 q^{36} -1.00000i q^{37} -2.93923i q^{38} +17.2782 q^{39} -8.89789 q^{41} -3.87847i q^{42} +5.61968i q^{43} +0.258786 q^{44} -8.51757 q^{46} +5.57834i q^{47} -2.93923i q^{48} +5.25879 q^{49} +12.5176 q^{51} -5.87847i q^{52} -12.8979i q^{53} -7.75694 q^{54} -1.31955 q^{56} +8.63910i q^{57} +3.61968i q^{58} -9.57834 q^{59} +0.380316 q^{61} +3.95865i q^{62} +7.44108i q^{63} -1.00000 q^{64} -0.760632 q^{66} +5.57834i q^{67} -4.25879i q^{68} +25.0351 q^{69} +11.2394 q^{71} +5.63910i q^{72} -2.51757i q^{73} +1.00000 q^{74} +2.93923 q^{76} +0.341481i q^{77} +17.2782i q^{78} +7.69987 q^{79} +5.88216 q^{81} -8.89789i q^{82} +6.17860i q^{83} +3.87847 q^{84} -5.61968 q^{86} -10.6391i q^{87} +0.258786i q^{88} -6.00000 q^{89} +7.75694 q^{91} -8.51757i q^{92} -11.6354i q^{93} -5.57834 q^{94} +2.93923 q^{96} -12.8979i q^{97} +5.25879i q^{98} +1.45932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56}+ \cdots - 38 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 - 22 * q^9 + 22 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^21 + 10 * q^29 + 6 * q^31 - 2 * q^34 + 22 * q^36 + 80 * q^39 - 18 * q^41 - 22 * q^44 - 4 * q^46 + 8 * q^49 + 28 * q^51 + 24 * q^54 - 2 * q^56 - 28 * q^59 + 14 * q^61 - 6 * q^64 - 28 * q^66 + 56 * q^69 + 44 * q^71 + 6 * q^74 + 52 * q^79 + 94 * q^81 - 12 * q^84 - 22 * q^86 - 36 * q^89 - 24 * q^91 - 4 * q^94 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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