LMFDB - Embedded newform 8450.2.a.bz.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,-1,3,0,-1,5,3,-4,0,-4,-1,0,5,0,3,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.24698 q^{6} +0.198062 q^{7} +1.00000 q^{8} -1.44504 q^{9} -1.10992 q^{11} +1.24698 q^{12} +0.198062 q^{14} +1.00000 q^{16} -6.49396 q^{17} -1.44504 q^{18} +0.493959 q^{19} +0.246980 q^{21} -1.10992 q^{22} +6.76271 q^{23} +1.24698 q^{24} -5.54288 q^{27} +0.198062 q^{28} -0.356896 q^{29} -6.27413 q^{31} +1.00000 q^{32} -1.38404 q^{33} -6.49396 q^{34} -1.44504 q^{36} +2.21983 q^{37} +0.493959 q^{38} -8.96615 q^{41} +0.246980 q^{42} -5.65279 q^{43} -1.10992 q^{44} +6.76271 q^{46} -1.43296 q^{47} +1.24698 q^{48} -6.96077 q^{49} -8.09783 q^{51} -9.92154 q^{53} -5.54288 q^{54} +0.198062 q^{56} +0.615957 q^{57} -0.356896 q^{58} -4.61596 q^{59} +5.50365 q^{61} -6.27413 q^{62} -0.286208 q^{63} +1.00000 q^{64} -1.38404 q^{66} -3.87263 q^{67} -6.49396 q^{68} +8.43296 q^{69} -13.6039 q^{71} -1.44504 q^{72} +0.591794 q^{73} +2.21983 q^{74} +0.493959 q^{76} -0.219833 q^{77} +2.93362 q^{79} -2.57673 q^{81} -8.96615 q^{82} +15.6625 q^{83} +0.246980 q^{84} -5.65279 q^{86} -0.445042 q^{87} -1.10992 q^{88} +17.0804 q^{89} +6.76271 q^{92} -7.82371 q^{93} -1.43296 q^{94} +1.24698 q^{96} +13.5797 q^{97} -6.96077 q^{98} +1.60388 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 5 q^{7} + 3 q^{8} - 4 q^{9} - 4 q^{11} - q^{12} + 5 q^{14} + 3 q^{16} - 10 q^{17} - 4 q^{18} - 8 q^{19} - 4 q^{21} - 4 q^{22} + 3 q^{23} - q^{24} + 2 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 5 * q^7 + 3 * q^8 - 4 * q^9 - 4 * q^11 - q^12 + 5 * q^14 + 3 * q^16 - 10 * q^17 - 4 * q^18 - 8 * q^19 - 4 * q^21 - 4 * q^22 + 3 * q^23 - q^24 + 2 * q^27 + 5 * q^28 + 3 * q^29 - 8 * q^31 + 3 * q^32 + 6 * q^33 - 10 * q^34 - 4 * q^36 + 8 * q^37 - 8 * q^38 - 11 * q^41 - 4 * q^42 + q^43 - 4 * q^44 + 3 * q^46 + 15 * q^47 - q^48 - 8 * q^49 - 6 * q^51 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 12 * q^57 + 3 * q^58 - 24 * q^59 - 15 * q^61 - 8 * q^62 - 9 * q^63 + 3 * q^64 + 6 * q^66 + 5 * q^67 - 10 * q^68 + 6 * q^69 - 32 * q^71 - 4 * q^72 - 26 * q^73 + 8 * q^74 - 8 * q^76 - 2 * q^77 + 2 * q^79 - 5 * q^81 - 11 * q^82 + 7 * q^83 - 4 * q^84 + q^86 - q^87 - 4 * q^88 + 17 * q^89 + 3 * q^92 - 16 * q^93 + 15 * q^94 - q^96 - 6 * q^97 - 8 * q^98 - 4 * 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
$3$	1.24698	0.719944	0.359972	−	0.932963i	$-0.382786\pi$
$3$	1.24698	0.719944	0.359972	+	0.932963i	$0.382786\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.24698	0.509077
$7$	0.198062	0.0748605	0.0374302	−	0.999299i	$-0.488083\pi$
$7$	0.198062	0.0748605	0.0374302	+	0.999299i	$0.488083\pi$
$8$	1.00000	0.353553
$9$	−1.44504	−0.481681
$10$	0	0
$11$	−1.10992	−0.334652	−0.167326	−	0.985902i	$-0.553513\pi$
$11$	−1.10992	−0.334652	−0.167326	+	0.985902i	$0.553513\pi$
$12$	1.24698	0.359972
$13$	0	0
$13$	0	0
$14$	0.198062	0.0529344
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.49396	−1.57502	−0.787508	−	0.616304i	$-0.788629\pi$
$17$	−6.49396	−1.57502	−0.787508	+	0.616304i	$0.788629\pi$
$18$	−1.44504	−0.340600
$19$	0.493959	0.113322	0.0566610	−	0.998393i	$-0.481955\pi$
$19$	0.493959	0.113322	0.0566610	+	0.998393i	$0.481955\pi$
$20$	0	0
$21$	0.246980	0.0538954
$22$	−1.10992	−0.236635
$23$	6.76271	1.41012	0.705061	−	0.709147i	$-0.250920\pi$
$23$	6.76271	1.41012	0.705061	+	0.709147i	$0.250920\pi$
$24$	1.24698	0.254539
$25$	0	0
$26$	0	0
$27$	−5.54288	−1.06673
$28$	0.198062	0.0374302
$29$	−0.356896	−0.0662739	−0.0331369	−	0.999451i	$-0.510550\pi$
$29$	−0.356896	−0.0662739	−0.0331369	+	0.999451i	$0.510550\pi$
$30$	0	0
$31$	−6.27413	−1.12687	−0.563433	−	0.826162i	$-0.690520\pi$
$31$	−6.27413	−1.12687	−0.563433	+	0.826162i	$0.690520\pi$
$32$	1.00000	0.176777
$33$	−1.38404	−0.240931
$34$	−6.49396	−1.11370
$35$	0	0
$36$	−1.44504	−0.240840
$37$	2.21983	0.364938	0.182469	−	0.983212i	$-0.441591\pi$
$37$	2.21983	0.364938	0.182469	+	0.983212i	$0.441591\pi$
$38$	0.493959	0.0801308
$39$	0	0
$40$	0	0
$41$	−8.96615	−1.40028	−0.700139	−	0.714007i	$-0.746878\pi$
$41$	−8.96615	−1.40028	−0.700139	+	0.714007i	$0.746878\pi$
$42$	0.246980	0.0381098
$43$	−5.65279	−0.862043	−0.431021	−	0.902342i	$-0.641847\pi$
$43$	−5.65279	−0.862043	−0.431021	+	0.902342i	$0.641847\pi$
$44$	−1.10992	−0.167326
$45$	0	0
$46$	6.76271	0.997107
$47$	−1.43296	−0.209019	−0.104509	−	0.994524i	$-0.533327\pi$
$47$	−1.43296	−0.209019	−0.104509	+	0.994524i	$0.533327\pi$
$48$	1.24698	0.179986
$49$	−6.96077	−0.994396
$50$	0	0
$51$	−8.09783	−1.13392
$52$	0	0
$53$	−9.92154	−1.36283	−0.681414	−	0.731898i	$-0.738635\pi$
$53$	−9.92154	−1.36283	−0.681414	+	0.731898i	$0.738635\pi$
$54$	−5.54288	−0.754290
$55$	0	0
$56$	0.198062	0.0264672
$57$	0.615957	0.0815855
$58$	−0.356896	−0.0468627
$59$	−4.61596	−0.600946	−0.300473	−	0.953790i	$-0.597145\pi$
$59$	−4.61596	−0.600946	−0.300473	+	0.953790i	$0.597145\pi$
$60$	0	0
$61$	5.50365	0.704670	0.352335	−	0.935874i	$-0.385388\pi$
$61$	5.50365	0.704670	0.352335	+	0.935874i	$0.385388\pi$
$62$	−6.27413	−0.796815
$63$	−0.286208	−0.0360589
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.38404	−0.170364
$67$	−3.87263	−0.473116	−0.236558	−	0.971617i	$-0.576019\pi$
$67$	−3.87263	−0.473116	−0.236558	+	0.971617i	$0.576019\pi$
$68$	−6.49396	−0.787508
$69$	8.43296	1.01521
$70$	0	0
$71$	−13.6039	−1.61448	−0.807241	−	0.590221i	$-0.799040\pi$
$71$	−13.6039	−1.61448	−0.807241	+	0.590221i	$0.799040\pi$
$72$	−1.44504	−0.170300
$73$	0.591794	0.0692642	0.0346321	−	0.999400i	$-0.488974\pi$
$73$	0.591794	0.0692642	0.0346321	+	0.999400i	$0.488974\pi$
$74$	2.21983	0.258050
$75$	0	0
$76$	0.493959	0.0566610
$77$	−0.219833	−0.0250522
$78$	0	0
$79$	2.93362	0.330059	0.165029	−	0.986289i	$-0.447228\pi$
$79$	2.93362	0.330059	0.165029	+	0.986289i	$0.447228\pi$
$80$	0	0
$81$	−2.57673	−0.286303
$82$	−8.96615	−0.990145
$83$	15.6625	1.71918	0.859590	−	0.510984i	$-0.170719\pi$
$83$	15.6625	1.71918	0.859590	+	0.510984i	$0.170719\pi$
$84$	0.246980	0.0269477
$85$	0	0
$86$	−5.65279	−0.609556
$87$	−0.445042	−0.0477135
$88$	−1.10992	−0.118317
$89$	17.0804	1.81052	0.905258	−	0.424862i	$-0.139677\pi$
$89$	17.0804	1.81052	0.905258	+	0.424862i	$0.139677\pi$
$90$	0	0
$91$	0	0
$92$	6.76271	0.705061
$93$	−7.82371	−0.811281
$94$	−1.43296	−0.147799
$95$	0	0
$96$	1.24698	0.127269
$97$	13.5797	1.37881	0.689405	−	0.724376i	$-0.257872\pi$
$97$	13.5797	1.37881	0.689405	+	0.724376i	$0.257872\pi$
$98$	−6.96077	−0.703144
$99$	1.60388	0.161196
$100$	0	0
$101$	−5.55496	−0.552739	−0.276369	−	0.961051i	$-0.589131\pi$
$101$	−5.55496	−0.552739	−0.276369	+	0.961051i	$0.589131\pi$
$102$	−8.09783	−0.801805
$103$	4.39373	0.432927	0.216464	−	0.976291i	$-0.430548\pi$
$103$	4.39373	0.432927	0.216464	+	0.976291i	$0.430548\pi$
$104$	0	0
$105$	0	0
$106$	−9.92154	−0.963665
$107$	17.9758	1.73779	0.868895	−	0.494997i	$-0.164831\pi$
$107$	17.9758	1.73779	0.868895	+	0.494997i	$0.164831\pi$
$108$	−5.54288	−0.533364
$109$	−6.73556	−0.645150	−0.322575	−	0.946544i	$-0.604548\pi$
$109$	−6.73556	−0.645150	−0.322575	+	0.946544i	$0.604548\pi$
$110$	0	0
$111$	2.76809	0.262735
$112$	0.198062	0.0187151
$113$	−2.27413	−0.213932	−0.106966	−	0.994263i	$-0.534114\pi$
$113$	−2.27413	−0.213932	−0.106966	+	0.994263i	$0.534114\pi$
$114$	0.615957	0.0576897
$115$	0	0
$116$	−0.356896	−0.0331369
$117$	0	0
$118$	−4.61596	−0.424933
$119$	−1.28621	−0.117907
$120$	0	0
$121$	−9.76809	−0.888008
$122$	5.50365	0.498277
$123$	−11.1806	−1.00812
$124$	−6.27413	−0.563433
$125$	0	0
$126$	−0.286208	−0.0254975
$127$	−13.7778	−1.22258	−0.611290	−	0.791407i	$-0.709349\pi$
$127$	−13.7778	−1.22258	−0.611290	+	0.791407i	$0.709349\pi$
$128$	1.00000	0.0883883
$129$	−7.04892	−0.620623
$130$	0	0
$131$	−19.3056	−1.68674	−0.843368	−	0.537336i	$-0.819431\pi$
$131$	−19.3056	−1.68674	−0.843368	+	0.537336i	$0.819431\pi$
$132$	−1.38404	−0.120465
$133$	0.0978347	0.00848334
$134$	−3.87263	−0.334544
$135$	0	0
$136$	−6.49396	−0.556852
$137$	19.5362	1.66909	0.834544	−	0.550941i	$-0.185731\pi$
$137$	19.5362	1.66909	0.834544	+	0.550941i	$0.185731\pi$
$138$	8.43296	0.717861
$139$	−18.4155	−1.56198	−0.780991	−	0.624542i	$-0.785286\pi$
$139$	−18.4155	−1.56198	−0.780991	+	0.624542i	$0.785286\pi$
$140$	0	0
$141$	−1.78687	−0.150482
$142$	−13.6039	−1.14161
$143$	0	0
$144$	−1.44504	−0.120420
$145$	0	0
$146$	0.591794	0.0489772
$147$	−8.67994	−0.715909
$148$	2.21983	0.182469
$149$	−17.2664	−1.41452	−0.707258	−	0.706956i	$-0.750068\pi$
$149$	−17.2664	−1.41452	−0.707258	+	0.706956i	$0.750068\pi$
$150$	0	0
$151$	−3.97584	−0.323549	−0.161775	−	0.986828i	$-0.551722\pi$
$151$	−3.97584	−0.323549	−0.161775	+	0.986828i	$0.551722\pi$
$152$	0.493959	0.0400654
$153$	9.38404	0.758655
$154$	−0.219833	−0.0177146
$155$	0	0
$156$	0	0
$157$	−17.4034	−1.38894	−0.694472	−	0.719520i	$-0.744362\pi$
$157$	−17.4034	−1.38894	−0.694472	+	0.719520i	$0.744362\pi$
$158$	2.93362	0.233387
$159$	−12.3720	−0.981160
$160$	0	0
$161$	1.33944	0.105562
$162$	−2.57673	−0.202447
$163$	−9.16852	−0.718134	−0.359067	−	0.933312i	$-0.616905\pi$
$163$	−9.16852	−0.718134	−0.359067	+	0.933312i	$0.616905\pi$
$164$	−8.96615	−0.700139
$165$	0	0
$166$	15.6625	1.21564
$167$	7.36227	0.569710	0.284855	−	0.958571i	$-0.408055\pi$
$167$	7.36227	0.569710	0.284855	+	0.958571i	$0.408055\pi$
$168$	0.246980	0.0190549
$169$	0	0
$170$	0	0
$171$	−0.713792	−0.0545850
$172$	−5.65279	−0.431021
$173$	8.59179	0.653222	0.326611	−	0.945159i	$-0.394093\pi$
$173$	8.59179	0.653222	0.326611	+	0.945159i	$0.394093\pi$
$174$	−0.445042	−0.0337385
$175$	0	0
$176$	−1.10992	−0.0836631
$177$	−5.75600	−0.432648
$178$	17.0804	1.28023
$179$	−0.659498	−0.0492932	−0.0246466	−	0.999696i	$-0.507846\pi$
$179$	−0.659498	−0.0492932	−0.0246466	+	0.999696i	$0.507846\pi$
$180$	0	0
$181$	−20.4101	−1.51707	−0.758536	−	0.651631i	$-0.774085\pi$
$181$	−20.4101	−1.51707	−0.758536	+	0.651631i	$0.774085\pi$
$182$	0	0
$183$	6.86294	0.507323
$184$	6.76271	0.498554
$185$	0	0
$186$	−7.82371	−0.573662
$187$	7.20775	0.527083
$188$	−1.43296	−0.104509
$189$	−1.09783	−0.0798557
$190$	0	0
$191$	12.7332	0.921340	0.460670	−	0.887572i	$-0.347609\pi$
$191$	12.7332	0.921340	0.460670	+	0.887572i	$0.347609\pi$
$192$	1.24698	0.0899930
$193$	8.68963	0.625493	0.312747	−	0.949837i	$-0.398751\pi$
$193$	8.68963	0.625493	0.312747	+	0.949837i	$0.398751\pi$
$194$	13.5797	0.974967
$195$	0	0
$196$	−6.96077	−0.497198
$197$	5.28621	0.376627	0.188313	−	0.982109i	$-0.439698\pi$
$197$	5.28621	0.376627	0.188313	+	0.982109i	$0.439698\pi$
$198$	1.60388	0.113982
$199$	−10.9444	−0.775826	−0.387913	−	0.921696i	$-0.626804\pi$
$199$	−10.9444	−0.775826	−0.387913	+	0.921696i	$0.626804\pi$
$200$	0	0
$201$	−4.82908	−0.340617
$202$	−5.55496	−0.390845
$203$	−0.0706876	−0.00496130
$204$	−8.09783	−0.566962
$205$	0	0
$206$	4.39373	0.306126
$207$	−9.77240	−0.679229
$208$	0	0
$209$	−0.548253	−0.0379235
$210$	0	0
$211$	3.30559	0.227566	0.113783	−	0.993506i	$-0.463703\pi$
$211$	3.30559	0.227566	0.113783	+	0.993506i	$0.463703\pi$
$212$	−9.92154	−0.681414
$213$	−16.9638	−1.16234
$214$	17.9758	1.22880
$215$	0	0
$216$	−5.54288	−0.377145
$217$	−1.24267	−0.0843578
$218$	−6.73556	−0.456190
$219$	0.737955	0.0498664
$220$	0	0
$221$	0	0
$222$	2.76809	0.185782
$223$	−16.6310	−1.11370	−0.556848	−	0.830615i	$-0.687989\pi$
$223$	−16.6310	−1.11370	−0.556848	+	0.830615i	$0.687989\pi$
$224$	0.198062	0.0132336
$225$	0	0
$226$	−2.27413	−0.151273
$227$	−10.8616	−0.720910	−0.360455	−	0.932777i	$-0.617379\pi$
$227$	−10.8616	−0.720910	−0.360455	+	0.932777i	$0.617379\pi$
$228$	0.615957	0.0407928
$229$	23.3991	1.54626	0.773128	−	0.634250i	$-0.218691\pi$
$229$	23.3991	1.54626	0.773128	+	0.634250i	$0.218691\pi$
$230$	0	0
$231$	−0.274127	−0.0180362
$232$	−0.356896	−0.0234314
$233$	−14.3913	−0.942808	−0.471404	−	0.881917i	$-0.656253\pi$
$233$	−14.3913	−0.942808	−0.471404	+	0.881917i	$0.656253\pi$
$234$	0	0
$235$	0	0
$236$	−4.61596	−0.300473
$237$	3.65817	0.237624
$238$	−1.28621	−0.0833725
$239$	−3.56033	−0.230299	−0.115149	−	0.993348i	$-0.536735\pi$
$239$	−3.56033	−0.230299	−0.115149	+	0.993348i	$0.536735\pi$
$240$	0	0
$241$	29.1890	1.88023	0.940113	−	0.340862i	$-0.110719\pi$
$241$	29.1890	1.88023	0.940113	+	0.340862i	$0.110719\pi$
$242$	−9.76809	−0.627916
$243$	13.4155	0.860605
$244$	5.50365	0.352335
$245$	0	0
$246$	−11.1806	−0.712849
$247$	0	0
$248$	−6.27413	−0.398407
$249$	19.5308	1.23771
$250$	0	0
$251$	−29.3599	−1.85318	−0.926590	−	0.376074i	$-0.877274\pi$
$251$	−29.3599	−1.85318	−0.926590	+	0.376074i	$0.877274\pi$
$252$	−0.286208	−0.0180294
$253$	−7.50604	−0.471901
$254$	−13.7778	−0.864494
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.5362	−1.34339	−0.671695	−	0.740828i	$-0.734433\pi$
$257$	−21.5362	−1.34339	−0.671695	+	0.740828i	$0.734433\pi$
$258$	−7.04892	−0.438846
$259$	0.439665	0.0273195
$260$	0	0
$261$	0.515729	0.0319229
$262$	−19.3056	−1.19270
$263$	6.57002	0.405125	0.202563	−	0.979269i	$-0.435073\pi$
$263$	6.57002	0.405125	0.202563	+	0.979269i	$0.435073\pi$
$264$	−1.38404	−0.0851820
$265$	0	0
$266$	0.0978347	0.00599863
$267$	21.2989	1.30347
$268$	−3.87263	−0.236558
$269$	7.36898	0.449294	0.224647	−	0.974440i	$-0.427877\pi$
$269$	7.36898	0.449294	0.224647	+	0.974440i	$0.427877\pi$
$270$	0	0
$271$	−15.5797	−0.946400	−0.473200	−	0.880955i	$-0.656901\pi$
$271$	−15.5797	−0.946400	−0.473200	+	0.880955i	$0.656901\pi$
$272$	−6.49396	−0.393754
$273$	0	0
$274$	19.5362	1.18022
$275$	0	0
$276$	8.43296	0.507605
$277$	13.9952	0.840891	0.420445	−	0.907318i	$-0.361874\pi$
$277$	13.9952	0.840891	0.420445	+	0.907318i	$0.361874\pi$
$278$	−18.4155	−1.10449
$279$	9.06638	0.542790
$280$	0	0
$281$	−13.3341	−0.795443	−0.397722	−	0.917506i	$-0.630199\pi$
$281$	−13.3341	−0.795443	−0.397722	+	0.917506i	$0.630199\pi$
$282$	−1.78687	−0.106407
$283$	20.7289	1.23220	0.616101	−	0.787667i	$-0.288711\pi$
$283$	20.7289	1.23220	0.616101	+	0.787667i	$0.288711\pi$
$284$	−13.6039	−0.807241
$285$	0	0
$286$	0	0
$287$	−1.77586	−0.104825
$288$	−1.44504	−0.0851499
$289$	25.1715	1.48068
$290$	0	0
$291$	16.9336	0.992667
$292$	0.591794	0.0346321
$293$	16.3913	0.957592	0.478796	−	0.877926i	$-0.341073\pi$
$293$	16.3913	0.957592	0.478796	+	0.877926i	$0.341073\pi$
$294$	−8.67994	−0.506224
$295$	0	0
$296$	2.21983	0.129025
$297$	6.15213	0.356983
$298$	−17.2664	−1.00021
$299$	0	0
$300$	0	0
$301$	−1.11960	−0.0645330
$302$	−3.97584	−0.228784
$303$	−6.92692	−0.397941
$304$	0.493959	0.0283305
$305$	0	0
$306$	9.38404	0.536450
$307$	−6.94869	−0.396583	−0.198291	−	0.980143i	$-0.563539\pi$
$307$	−6.94869	−0.396583	−0.198291	+	0.980143i	$0.563539\pi$
$308$	−0.219833	−0.0125261
$309$	5.47889	0.311683
$310$	0	0
$311$	−24.3612	−1.38140	−0.690699	−	0.723143i	$-0.742697\pi$
$311$	−24.3612	−1.38140	−0.690699	+	0.723143i	$0.742697\pi$
$312$	0	0
$313$	−2.89008	−0.163357	−0.0816786	−	0.996659i	$-0.526028\pi$
$313$	−2.89008	−0.163357	−0.0816786	+	0.996659i	$0.526028\pi$
$314$	−17.4034	−0.982132
$315$	0	0
$316$	2.93362	0.165029
$317$	5.01208	0.281507	0.140753	−	0.990045i	$-0.455048\pi$
$317$	5.01208	0.281507	0.140753	+	0.990045i	$0.455048\pi$
$318$	−12.3720	−0.693785
$319$	0.396125	0.0221787
$320$	0	0
$321$	22.4155	1.25111
$322$	1.33944	0.0746439
$323$	−3.20775	−0.178484
$324$	−2.57673	−0.143152
$325$	0	0
$326$	−9.16852	−0.507797
$327$	−8.39911	−0.464472
$328$	−8.96615	−0.495073
$329$	−0.283815	−0.0156472
$330$	0	0
$331$	24.0194	1.32022	0.660112	−	0.751167i	$-0.270509\pi$
$331$	24.0194	1.32022	0.660112	+	0.751167i	$0.270509\pi$
$332$	15.6625	0.859590
$333$	−3.20775	−0.175784
$334$	7.36227	0.402846
$335$	0	0
$336$	0.246980	0.0134738
$337$	−28.5133	−1.55322	−0.776610	−	0.629981i	$-0.783062\pi$
$337$	−28.5133	−1.55322	−0.776610	+	0.629981i	$0.783062\pi$
$338$	0	0
$339$	−2.83579	−0.154019
$340$	0	0
$341$	6.96376	0.377108
$342$	−0.713792	−0.0385974
$343$	−2.76510	−0.149301
$344$	−5.65279	−0.304778
$345$	0	0
$346$	8.59179	0.461898
$347$	25.1987	1.35273	0.676367	−	0.736565i	$-0.263553\pi$
$347$	25.1987	1.35273	0.676367	+	0.736565i	$0.263553\pi$
$348$	−0.445042	−0.0238567
$349$	16.4155	0.878702	0.439351	−	0.898316i	$-0.355209\pi$
$349$	16.4155	0.878702	0.439351	+	0.898316i	$0.355209\pi$
$350$	0	0
$351$	0	0
$352$	−1.10992	−0.0591587
$353$	−1.10992	−0.0590749	−0.0295374	−	0.999564i	$-0.509403\pi$
$353$	−1.10992	−0.0590749	−0.0295374	+	0.999564i	$0.509403\pi$
$354$	−5.75600	−0.305928
$355$	0	0
$356$	17.0804	0.905258
$357$	−1.60388	−0.0848861
$358$	−0.659498	−0.0348555
$359$	36.7633	1.94029	0.970146	−	0.242520i	$-0.0779740\pi$
$359$	36.7633	1.94029	0.970146	+	0.242520i	$0.0779740\pi$
$360$	0	0
$361$	−18.7560	−0.987158
$362$	−20.4101	−1.07273
$363$	−12.1806	−0.639316
$364$	0	0
$365$	0	0
$366$	6.86294	0.358731
$367$	−7.09485	−0.370348	−0.185174	−	0.982706i	$-0.559285\pi$
$367$	−7.09485	−0.370348	−0.185174	+	0.982706i	$0.559285\pi$
$368$	6.76271	0.352531
$369$	12.9565	0.674486
$370$	0	0
$371$	−1.96508	−0.102022
$372$	−7.82371	−0.405640
$373$	−26.8901	−1.39232	−0.696158	−	0.717889i	$-0.745109\pi$
$373$	−26.8901	−1.39232	−0.696158	+	0.717889i	$0.745109\pi$
$374$	7.20775	0.372704
$375$	0	0
$376$	−1.43296	−0.0738993
$377$	0	0
$378$	−1.09783	−0.0564665
$379$	8.46980	0.435064	0.217532	−	0.976053i	$-0.430199\pi$
$379$	8.46980	0.435064	0.217532	+	0.976053i	$0.430199\pi$
$380$	0	0
$381$	−17.1806	−0.880189
$382$	12.7332	0.651486
$383$	4.79656	0.245093	0.122546	−	0.992463i	$-0.460894\pi$
$383$	4.79656	0.245093	0.122546	+	0.992463i	$0.460894\pi$
$384$	1.24698	0.0636347
$385$	0	0
$386$	8.68963	0.442290
$387$	8.16852	0.415229
$388$	13.5797	0.689405
$389$	29.2131	1.48116	0.740582	−	0.671966i	$-0.234550\pi$
$389$	29.2131	1.48116	0.740582	+	0.671966i	$0.234550\pi$
$390$	0	0
$391$	−43.9168	−2.22097
$392$	−6.96077	−0.351572
$393$	−24.0737	−1.21436
$394$	5.28621	0.266315
$395$	0	0
$396$	1.60388	0.0805978
$397$	22.6112	1.13482	0.567411	−	0.823435i	$-0.307945\pi$
$397$	22.6112	1.13482	0.567411	+	0.823435i	$0.307945\pi$
$398$	−10.9444	−0.548592
$399$	0.121998	0.00610753
$400$	0	0
$401$	10.5894	0.528809	0.264405	−	0.964412i	$-0.414825\pi$
$401$	10.5894	0.528809	0.264405	+	0.964412i	$0.414825\pi$
$402$	−4.82908	−0.240853
$403$	0	0
$404$	−5.55496	−0.276369
$405$	0	0
$406$	−0.0706876	−0.00350817
$407$	−2.46383	−0.122127
$408$	−8.09783	−0.400903
$409$	11.7127	0.579157	0.289579	−	0.957154i	$-0.406485\pi$
$409$	11.7127	0.579157	0.289579	+	0.957154i	$0.406485\pi$
$410$	0	0
$411$	24.3612	1.20165
$412$	4.39373	0.216464
$413$	−0.914247	−0.0449871
$414$	−9.77240	−0.480287
$415$	0	0
$416$	0	0
$417$	−22.9638	−1.12454
$418$	−0.548253	−0.0268159
$419$	21.3250	1.04179	0.520896	−	0.853620i	$-0.325598\pi$
$419$	21.3250	1.04179	0.520896	+	0.853620i	$0.325598\pi$
$420$	0	0
$421$	22.1457	1.07931	0.539657	−	0.841885i	$-0.318554\pi$
$421$	22.1457	1.07931	0.539657	+	0.841885i	$0.318554\pi$
$422$	3.30559	0.160913
$423$	2.07069	0.100680
$424$	−9.92154	−0.481833
$425$	0	0
$426$	−16.9638	−0.821897
$427$	1.09006	0.0527519
$428$	17.9758	0.868895
$429$	0	0
$430$	0	0
$431$	−36.8068	−1.77292	−0.886462	−	0.462802i	$-0.846844\pi$
$431$	−36.8068	−1.77292	−0.886462	+	0.462802i	$0.846844\pi$
$432$	−5.54288	−0.266682
$433$	−2.08708	−0.100299	−0.0501494	−	0.998742i	$-0.515970\pi$
$433$	−2.08708	−0.100299	−0.0501494	+	0.998742i	$0.515970\pi$
$434$	−1.24267	−0.0596500
$435$	0	0
$436$	−6.73556	−0.322575
$437$	3.34050	0.159798
$438$	0.737955	0.0352608
$439$	9.95646	0.475196	0.237598	−	0.971364i	$-0.423640\pi$
$439$	9.95646	0.475196	0.237598	+	0.971364i	$0.423640\pi$
$440$	0	0
$441$	10.0586	0.478981
$442$	0	0
$443$	15.3884	0.731123	0.365561	−	0.930787i	$-0.380877\pi$
$443$	15.3884	0.731123	0.365561	+	0.930787i	$0.380877\pi$
$444$	2.76809	0.131368
$445$	0	0
$446$	−16.6310	−0.787502
$447$	−21.5308	−1.01837
$448$	0.198062	0.00935756
$449$	−8.18060	−0.386067	−0.193033	−	0.981192i	$-0.561833\pi$
$449$	−8.18060	−0.386067	−0.193033	+	0.981192i	$0.561833\pi$
$450$	0	0
$451$	9.95167	0.468606
$452$	−2.27413	−0.106966
$453$	−4.95779	−0.232937
$454$	−10.8616	−0.509761
$455$	0	0
$456$	0.615957	0.0288448
$457$	16.0737	0.751895	0.375947	−	0.926641i	$-0.377317\pi$
$457$	16.0737	0.751895	0.375947	+	0.926641i	$0.377317\pi$
$458$	23.3991	1.09337
$459$	35.9952	1.68011
$460$	0	0
$461$	−27.5308	−1.28224	−0.641118	−	0.767442i	$-0.721529\pi$
$461$	−27.5308	−1.28224	−0.641118	+	0.767442i	$0.721529\pi$
$462$	−0.274127	−0.0127535
$463$	−10.6595	−0.495389	−0.247694	−	0.968838i	$-0.579673\pi$
$463$	−10.6595	−0.495389	−0.247694	+	0.968838i	$0.579673\pi$
$464$	−0.356896	−0.0165685
$465$	0	0
$466$	−14.3913	−0.666666
$467$	10.7976	0.499655	0.249827	−	0.968290i	$-0.419626\pi$
$467$	10.7976	0.499655	0.249827	+	0.968290i	$0.419626\pi$
$468$	0	0
$469$	−0.767021	−0.0354177
$470$	0	0
$471$	−21.7017	−0.999962
$472$	−4.61596	−0.212467
$473$	6.27413	0.288485
$474$	3.65817	0.168025
$475$	0	0
$476$	−1.28621	−0.0589533
$477$	14.3370	0.656448
$478$	−3.56033	−0.162846
$479$	−9.08575	−0.415139	−0.207569	−	0.978220i	$-0.566555\pi$
$479$	−9.08575	−0.415139	−0.207569	+	0.978220i	$0.566555\pi$
$480$	0	0
$481$	0	0
$482$	29.1890	1.32952
$483$	1.67025	0.0759991
$484$	−9.76809	−0.444004
$485$	0	0
$486$	13.4155	0.608540
$487$	−40.8471	−1.85096	−0.925480	−	0.378796i	$-0.876338\pi$
$487$	−40.8471	−1.85096	−0.925480	+	0.378796i	$0.876338\pi$
$488$	5.50365	0.249138
$489$	−11.4330	−0.517016
$490$	0	0
$491$	29.0944	1.31301	0.656505	−	0.754321i	$-0.272034\pi$
$491$	29.0944	1.31301	0.656505	+	0.754321i	$0.272034\pi$
$492$	−11.1806	−0.504061
$493$	2.31767	0.104382
$494$	0	0
$495$	0	0
$496$	−6.27413	−0.281717
$497$	−2.69441	−0.120861
$498$	19.5308	0.875196
$499$	−16.2150	−0.725885	−0.362943	−	0.931812i	$-0.618228\pi$
$499$	−16.2150	−0.725885	−0.362943	+	0.931812i	$0.618228\pi$
$500$	0	0
$501$	9.18060	0.410159
$502$	−29.3599	−1.31040
$503$	1.49502	0.0666598	0.0333299	−	0.999444i	$-0.489389\pi$
$503$	1.49502	0.0666598	0.0333299	+	0.999444i	$0.489389\pi$
$504$	−0.286208	−0.0127487
$505$	0	0
$506$	−7.50604	−0.333684
$507$	0	0
$508$	−13.7778	−0.611290
$509$	4.63102	0.205266	0.102633	−	0.994719i	$-0.467273\pi$
$509$	4.63102	0.205266	0.102633	+	0.994719i	$0.467273\pi$
$510$	0	0
$511$	0.117212	0.00518516
$512$	1.00000	0.0441942
$513$	−2.73795	−0.120884
$514$	−21.5362	−0.949920
$515$	0	0
$516$	−7.04892	−0.310311
$517$	1.59047	0.0699486
$518$	0.439665	0.0193178
$519$	10.7138	0.470283
$520$	0	0
$521$	−9.23729	−0.404693	−0.202347	−	0.979314i	$-0.564857\pi$
$521$	−9.23729	−0.404693	−0.202347	+	0.979314i	$0.564857\pi$
$522$	0.515729	0.0225729
$523$	2.88471	0.126139	0.0630697	−	0.998009i	$-0.479911\pi$
$523$	2.88471	0.126139	0.0630697	+	0.998009i	$0.479911\pi$
$524$	−19.3056	−0.843368
$525$	0	0
$526$	6.57002	0.286467
$527$	40.7439	1.77483
$528$	−1.38404	−0.0602327
$529$	22.7342	0.988445
$530$	0	0
$531$	6.67025	0.289464
$532$	0.0978347	0.00424167
$533$	0	0
$534$	21.2989	0.921693
$535$	0	0
$536$	−3.87263	−0.167272
$537$	−0.822380	−0.0354883
$538$	7.36898	0.317699
$539$	7.72587	0.332777
$540$	0	0
$541$	−28.2301	−1.21371	−0.606854	−	0.794814i	$-0.707569\pi$
$541$	−28.2301	−1.21371	−0.606854	+	0.794814i	$0.707569\pi$
$542$	−15.5797	−0.669206
$543$	−25.4510	−1.09221
$544$	−6.49396	−0.278426
$545$	0	0
$546$	0	0
$547$	−37.8756	−1.61944	−0.809722	−	0.586814i	$-0.800382\pi$
$547$	−37.8756	−1.61944	−0.809722	+	0.586814i	$0.800382\pi$
$548$	19.5362	0.834544
$549$	−7.95300	−0.339426
$550$	0	0
$551$	−0.176292	−0.00751029
$552$	8.43296	0.358931
$553$	0.581040	0.0247083
$554$	13.9952	0.594600
$555$	0	0
$556$	−18.4155	−0.780991
$557$	−41.2030	−1.74583	−0.872913	−	0.487876i	$-0.837772\pi$
$557$	−41.2030	−1.74583	−0.872913	+	0.487876i	$0.837772\pi$
$558$	9.06638	0.383810
$559$	0	0
$560$	0	0
$561$	8.98792	0.379470
$562$	−13.3341	−0.562463
$563$	−8.00836	−0.337512	−0.168756	−	0.985658i	$-0.553975\pi$
$563$	−8.00836	−0.337512	−0.168756	+	0.985658i	$0.553975\pi$
$564$	−1.78687	−0.0752409
$565$	0	0
$566$	20.7289	0.871299
$567$	−0.510353	−0.0214328
$568$	−13.6039	−0.570806
$569$	43.1377	1.80842	0.904212	−	0.427083i	$-0.140459\pi$
$569$	43.1377	1.80842	0.904212	+	0.427083i	$0.140459\pi$
$570$	0	0
$571$	−8.98792	−0.376133	−0.188066	−	0.982156i	$-0.560222\pi$
$571$	−8.98792	−0.376133	−0.188066	+	0.982156i	$0.560222\pi$
$572$	0	0
$573$	15.8780	0.663313
$574$	−1.77586	−0.0741228
$575$	0	0
$576$	−1.44504	−0.0602101
$577$	1.03624	0.0431394	0.0215697	−	0.999767i	$-0.493134\pi$
$577$	1.03624	0.0431394	0.0215697	+	0.999767i	$0.493134\pi$
$578$	25.1715	1.04700
$579$	10.8358	0.450320
$580$	0	0
$581$	3.10215	0.128699
$582$	16.9336	0.701921
$583$	11.0121	0.456074
$584$	0.591794	0.0244886
$585$	0	0
$586$	16.3913	0.677120
$587$	3.70410	0.152885	0.0764423	−	0.997074i	$-0.475644\pi$
$587$	3.70410	0.152885	0.0764423	+	0.997074i	$0.475644\pi$
$588$	−8.67994	−0.357955
$589$	−3.09916	−0.127699
$590$	0	0
$591$	6.59179	0.271150
$592$	2.21983	0.0912346
$593$	−12.2064	−0.501258	−0.250629	−	0.968083i	$-0.580637\pi$
$593$	−12.2064	−0.501258	−0.250629	+	0.968083i	$0.580637\pi$
$594$	6.15213	0.252425
$595$	0	0
$596$	−17.2664	−0.707258
$597$	−13.6474	−0.558552
$598$	0	0
$599$	−31.5013	−1.28711	−0.643553	−	0.765401i	$-0.722540\pi$
$599$	−31.5013	−1.28711	−0.643553	+	0.765401i	$0.722540\pi$
$600$	0	0
$601$	19.6907	0.803200	0.401600	−	0.915815i	$-0.368454\pi$
$601$	19.6907	0.803200	0.401600	+	0.915815i	$0.368454\pi$
$602$	−1.11960	−0.0456317
$603$	5.59611	0.227891
$604$	−3.97584	−0.161775
$605$	0	0
$606$	−6.92692	−0.281387
$607$	31.0562	1.26053	0.630266	−	0.776379i	$-0.282946\pi$
$607$	31.0562	1.26053	0.630266	+	0.776379i	$0.282946\pi$
$608$	0.493959	0.0200327
$609$	−0.0881460	−0.00357186
$610$	0	0
$611$	0	0
$612$	9.38404	0.379327
$613$	−18.3720	−0.742037	−0.371018	−	0.928626i	$-0.620991\pi$
$613$	−18.3720	−0.742037	−0.371018	+	0.928626i	$0.620991\pi$
$614$	−6.94869	−0.280426
$615$	0	0
$616$	−0.219833	−0.00885730
$617$	9.27545	0.373416	0.186708	−	0.982415i	$-0.440218\pi$
$617$	9.27545	0.373416	0.186708	+	0.982415i	$0.440218\pi$
$618$	5.47889	0.220393
$619$	−14.9095	−0.599262	−0.299631	−	0.954055i	$-0.596864\pi$
$619$	−14.9095	−0.599262	−0.299631	+	0.954055i	$0.596864\pi$
$620$	0	0
$621$	−37.4849	−1.50422
$622$	−24.3612	−0.976795
$623$	3.38298	0.135536
$624$	0	0
$625$	0	0
$626$	−2.89008	−0.115511
$627$	−0.683661	−0.0273028
$628$	−17.4034	−0.694472
$629$	−14.4155	−0.574784
$630$	0	0
$631$	30.0823	1.19756	0.598779	−	0.800915i	$-0.295653\pi$
$631$	30.0823	1.19756	0.598779	+	0.800915i	$0.295653\pi$
$632$	2.93362	0.116693
$633$	4.12200	0.163835
$634$	5.01208	0.199055
$635$	0	0
$636$	−12.3720	−0.490580
$637$	0	0
$638$	0.396125	0.0156827
$639$	19.6582	0.777665
$640$	0	0
$641$	12.1787	0.481029	0.240515	−	0.970646i	$-0.422684\pi$
$641$	12.1787	0.481029	0.240515	+	0.970646i	$0.422684\pi$
$642$	22.4155	0.884669
$643$	3.36467	0.132689	0.0663447	−	0.997797i	$-0.478866\pi$
$643$	3.36467	0.132689	0.0663447	+	0.997797i	$0.478866\pi$
$644$	1.33944	0.0527812
$645$	0	0
$646$	−3.20775	−0.126207
$647$	−28.3913	−1.11618	−0.558089	−	0.829781i	$-0.688465\pi$
$647$	−28.3913	−1.11618	−0.558089	+	0.829781i	$0.688465\pi$
$648$	−2.57673	−0.101223
$649$	5.12333	0.201108
$650$	0	0
$651$	−1.54958	−0.0607329
$652$	−9.16852	−0.359067
$653$	25.6823	1.00503	0.502514	−	0.864569i	$-0.332409\pi$
$653$	25.6823	1.00503	0.502514	+	0.864569i	$0.332409\pi$
$654$	−8.39911	−0.328431
$655$	0	0
$656$	−8.96615	−0.350069
$657$	−0.855167	−0.0333632
$658$	−0.283815	−0.0110643
$659$	−5.26205	−0.204980	−0.102490	−	0.994734i	$-0.532681\pi$
$659$	−5.26205	−0.204980	−0.102490	+	0.994734i	$0.532681\pi$
$660$	0	0
$661$	−32.9476	−1.28151	−0.640757	−	0.767744i	$-0.721379\pi$
$661$	−32.9476	−1.28151	−0.640757	+	0.767744i	$0.721379\pi$
$662$	24.0194	0.933540
$663$	0	0
$664$	15.6625	0.607822
$665$	0	0
$666$	−3.20775	−0.124298
$667$	−2.41358	−0.0934543
$668$	7.36227	0.284855
$669$	−20.7385	−0.801799
$670$	0	0
$671$	−6.10859	−0.235819
$672$	0.246980	0.00952745
$673$	−20.1655	−0.777324	−0.388662	−	0.921380i	$-0.627063\pi$
$673$	−20.1655	−0.777324	−0.388662	+	0.921380i	$0.627063\pi$
$674$	−28.5133	−1.09829
$675$	0	0
$676$	0	0
$677$	−46.4590	−1.78557	−0.892783	−	0.450487i	$-0.851250\pi$
$677$	−46.4590	−1.78557	−0.892783	+	0.450487i	$0.851250\pi$
$678$	−2.83579	−0.108908
$679$	2.68963	0.103218
$680$	0	0
$681$	−13.5442	−0.519015
$682$	6.96376	0.266656
$683$	27.0911	1.03661	0.518307	−	0.855195i	$-0.326563\pi$
$683$	27.0911	1.03661	0.518307	+	0.855195i	$0.326563\pi$
$684$	−0.713792	−0.0272925
$685$	0	0
$686$	−2.76510	−0.105572
$687$	29.1782	1.11322
$688$	−5.65279	−0.215511
$689$	0	0
$690$	0	0
$691$	−3.47112	−0.132048	−0.0660239	−	0.997818i	$-0.521031\pi$
$691$	−3.47112	−0.132048	−0.0660239	+	0.997818i	$0.521031\pi$
$692$	8.59179	0.326611
$693$	0.317667	0.0120672
$694$	25.1987	0.956528
$695$	0	0
$696$	−0.445042	−0.0168693
$697$	58.2258	2.20546
$698$	16.4155	0.621336
$699$	−17.9457	−0.678769
$700$	0	0
$701$	8.07606	0.305029	0.152514	−	0.988301i	$-0.451263\pi$
$701$	8.07606	0.305029	0.152514	+	0.988301i	$0.451263\pi$
$702$	0	0
$703$	1.09651	0.0413555
$704$	−1.10992	−0.0418315
$705$	0	0
$706$	−1.10992	−0.0417722
$707$	−1.10023	−0.0413783
$708$	−5.75600	−0.216324
$709$	20.5157	0.770484	0.385242	−	0.922816i	$-0.374118\pi$
$709$	20.5157	0.770484	0.385242	+	0.922816i	$0.374118\pi$
$710$	0	0
$711$	−4.23921	−0.158983
$712$	17.0804	0.640114
$713$	−42.4301	−1.58902
$714$	−1.60388	−0.0600235
$715$	0	0
$716$	−0.659498	−0.0246466
$717$	−4.43967	−0.165802
$718$	36.7633	1.37199
$719$	52.4650	1.95661	0.978307	−	0.207159i	$-0.0664216\pi$
$719$	52.4650	1.95661	0.978307	+	0.207159i	$0.0664216\pi$
$720$	0	0
$721$	0.870232	0.0324091
$722$	−18.7560	−0.698026
$723$	36.3980	1.35366
$724$	−20.4101	−0.758536
$725$	0	0
$726$	−12.1806	−0.452065
$727$	−4.11397	−0.152579	−0.0762893	−	0.997086i	$-0.524307\pi$
$727$	−4.11397	−0.152579	−0.0762893	+	0.997086i	$0.524307\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	0	0
$731$	36.7090	1.35773
$732$	6.86294	0.253661
$733$	−20.9336	−0.773201	−0.386601	−	0.922247i	$-0.626351\pi$
$733$	−20.9336	−0.773201	−0.386601	+	0.922247i	$0.626351\pi$
$734$	−7.09485	−0.261876
$735$	0	0
$736$	6.76271	0.249277
$737$	4.29829	0.158330
$738$	12.9565	0.476934
$739$	25.8297	0.950160	0.475080	−	0.879943i	$-0.342419\pi$
$739$	25.8297	0.950160	0.475080	+	0.879943i	$0.342419\pi$
$740$	0	0
$741$	0	0
$742$	−1.96508	−0.0721405
$743$	52.0411	1.90920	0.954602	−	0.297883i	$-0.0962807\pi$
$743$	52.0411	1.90920	0.954602	+	0.297883i	$0.0962807\pi$
$744$	−7.82371	−0.286831
$745$	0	0
$746$	−26.8901	−0.984516
$747$	−22.6329	−0.828096
$748$	7.20775	0.263541
$749$	3.56033	0.130092
$750$	0	0
$751$	7.89738	0.288179	0.144090	−	0.989565i	$-0.453975\pi$
$751$	7.89738	0.288179	0.144090	+	0.989565i	$0.453975\pi$
$752$	−1.43296	−0.0522547
$753$	−36.6112	−1.33419
$754$	0	0
$755$	0	0
$756$	−1.09783	−0.0399279
$757$	16.1366	0.586494	0.293247	−	0.956037i	$-0.405264\pi$
$757$	16.1366	0.586494	0.293247	+	0.956037i	$0.405264\pi$
$758$	8.46980	0.307637
$759$	−9.35988	−0.339742
$760$	0	0
$761$	32.4064	1.17473	0.587366	−	0.809322i	$-0.300165\pi$
$761$	32.4064	1.17473	0.587366	+	0.809322i	$0.300165\pi$
$762$	−17.1806	−0.622388
$763$	−1.33406	−0.0482962
$764$	12.7332	0.460670
$765$	0	0
$766$	4.79656	0.173307
$767$	0	0
$768$	1.24698	0.0449965
$769$	5.44935	0.196509	0.0982544	−	0.995161i	$-0.468674\pi$
$769$	5.44935	0.196509	0.0982544	+	0.995161i	$0.468674\pi$
$770$	0	0
$771$	−26.8552	−0.967165
$772$	8.68963	0.312747
$773$	35.0616	1.26108	0.630539	−	0.776158i	$-0.282834\pi$
$773$	35.0616	1.26108	0.630539	+	0.776158i	$0.282834\pi$
$774$	8.16852	0.293611
$775$	0	0
$776$	13.5797	0.487483
$777$	0.548253	0.0196685
$778$	29.2131	1.04734
$779$	−4.42891	−0.158682
$780$	0	0
$781$	15.0992	0.540291
$782$	−43.9168	−1.57046
$783$	1.97823	0.0706962
$784$	−6.96077	−0.248599
$785$	0	0
$786$	−24.0737	−0.858679
$787$	−40.2344	−1.43420	−0.717101	−	0.696969i	$-0.754531\pi$
$787$	−40.2344	−1.43420	−0.717101	+	0.696969i	$0.754531\pi$
$788$	5.28621	0.188313
$789$	8.19269	0.291667
$790$	0	0
$791$	−0.450419	−0.0160150
$792$	1.60388	0.0569912
$793$	0	0
$794$	22.6112	0.802440
$795$	0	0
$796$	−10.9444	−0.387913
$797$	2.73317	0.0968138	0.0484069	−	0.998828i	$-0.484586\pi$
$797$	2.73317	0.0968138	0.0484069	+	0.998828i	$0.484586\pi$
$798$	0.121998	0.00431868
$799$	9.30559	0.329208
$800$	0	0
$801$	−24.6819	−0.872091
$802$	10.5894	0.373925
$803$	−0.656842	−0.0231794
$804$	−4.82908	−0.170309
$805$	0	0
$806$	0	0
$807$	9.18896	0.323467
$808$	−5.55496	−0.195423
$809$	−36.4782	−1.28250	−0.641252	−	0.767330i	$-0.721585\pi$
$809$	−36.4782	−1.28250	−0.641252	+	0.767330i	$0.721585\pi$
$810$	0	0
$811$	3.51679	0.123491	0.0617457	−	0.998092i	$-0.480333\pi$
$811$	3.51679	0.123491	0.0617457	+	0.998092i	$0.480333\pi$
$812$	−0.0706876	−0.00248065
$813$	−19.4276	−0.681355
$814$	−2.46383	−0.0863571
$815$	0	0
$816$	−8.09783	−0.283481
$817$	−2.79225	−0.0976884
$818$	11.7127	0.409526
$819$	0	0
$820$	0	0
$821$	29.3980	1.02600	0.512999	−	0.858389i	$-0.328534\pi$
$821$	29.3980	1.02600	0.512999	+	0.858389i	$0.328534\pi$
$822$	24.3612	0.849695
$823$	−53.7318	−1.87297	−0.936487	−	0.350702i	$-0.885943\pi$
$823$	−53.7318	−1.87297	−0.936487	+	0.350702i	$0.885943\pi$
$824$	4.39373	0.153063
$825$	0	0
$826$	−0.914247	−0.0318107
$827$	47.3099	1.64513	0.822563	−	0.568674i	$-0.192543\pi$
$827$	47.3099	1.64513	0.822563	+	0.568674i	$0.192543\pi$
$828$	−9.77240	−0.339614
$829$	−49.3749	−1.71486	−0.857431	−	0.514598i	$-0.827941\pi$
$829$	−49.3749	−1.71486	−0.857431	+	0.514598i	$0.827941\pi$
$830$	0	0
$831$	17.4517	0.605394
$832$	0	0
$833$	45.2030	1.56619
$834$	−22.9638	−0.795170
$835$	0	0
$836$	−0.548253	−0.0189617
$837$	34.7767	1.20206
$838$	21.3250	0.736659
$839$	10.6219	0.366710	0.183355	−	0.983047i	$-0.441304\pi$
$839$	10.6219	0.366710	0.183355	+	0.983047i	$0.441304\pi$
$840$	0	0
$841$	−28.8726	−0.995608
$842$	22.1457	0.763191
$843$	−16.6273	−0.572675
$844$	3.30559	0.113783
$845$	0	0
$846$	2.07069	0.0711917
$847$	−1.93469	−0.0664767
$848$	−9.92154	−0.340707
$849$	25.8485	0.887117
$850$	0	0
$851$	15.0121	0.514608
$852$	−16.9638	−0.581169
$853$	−11.4625	−0.392469	−0.196234	−	0.980557i	$-0.562871\pi$
$853$	−11.4625	−0.392469	−0.196234	+	0.980557i	$0.562871\pi$
$854$	1.09006	0.0373013
$855$	0	0
$856$	17.9758	0.614401
$857$	19.4819	0.665488	0.332744	−	0.943017i	$-0.392025\pi$
$857$	19.4819	0.665488	0.332744	+	0.943017i	$0.392025\pi$
$858$	0	0
$859$	−23.1448	−0.789692	−0.394846	−	0.918747i	$-0.629202\pi$
$859$	−23.1448	−0.789692	−0.394846	+	0.918747i	$0.629202\pi$
$860$	0	0
$861$	−2.21446	−0.0754684
$862$	−36.8068	−1.25365
$863$	38.9681	1.32649	0.663244	−	0.748403i	$-0.269179\pi$
$863$	38.9681	1.32649	0.663244	+	0.748403i	$0.269179\pi$
$864$	−5.54288	−0.188572
$865$	0	0
$866$	−2.08708	−0.0709219
$867$	31.3884	1.06600
$868$	−1.24267	−0.0421789
$869$	−3.25608	−0.110455
$870$	0	0
$871$	0	0
$872$	−6.73556	−0.228095
$873$	−19.6233	−0.664146
$874$	3.34050	0.112994
$875$	0	0
$876$	0.737955	0.0249332
$877$	28.2064	0.952463	0.476232	−	0.879320i	$-0.342002\pi$
$877$	28.2064	0.952463	0.476232	+	0.879320i	$0.342002\pi$
$878$	9.95646	0.336014
$879$	20.4397	0.689413
$880$	0	0
$881$	−21.2948	−0.717441	−0.358721	−	0.933445i	$-0.616787\pi$
$881$	−21.2948	−0.717441	−0.358721	+	0.933445i	$0.616787\pi$
$882$	10.0586	0.338691
$883$	46.8068	1.57518	0.787588	−	0.616202i	$-0.211330\pi$
$883$	46.8068	1.57518	0.787588	+	0.616202i	$0.211330\pi$
$884$	0	0
$885$	0	0
$886$	15.3884	0.516982
$887$	−1.62565	−0.0545838	−0.0272919	−	0.999628i	$-0.508688\pi$
$887$	−1.62565	−0.0545838	−0.0272919	+	0.999628i	$0.508688\pi$
$888$	2.76809	0.0928909
$889$	−2.72886	−0.0915229
$890$	0	0
$891$	2.85995	0.0958120
$892$	−16.6310	−0.556848
$893$	−0.707824	−0.0236864
$894$	−21.5308	−0.720098
$895$	0	0
$896$	0.198062	0.00661680
$897$	0	0
$898$	−8.18060	−0.272990
$899$	2.23921	0.0746818
$900$	0	0
$901$	64.4301	2.14648
$902$	9.95167	0.331354
$903$	−1.39612	−0.0464601
$904$	−2.27413	−0.0756363
$905$	0	0
$906$	−4.95779	−0.164711
$907$	28.8442	0.957754	0.478877	−	0.877882i	$-0.341044\pi$
$907$	28.8442	0.957754	0.478877	+	0.877882i	$0.341044\pi$
$908$	−10.8616	−0.360455
$909$	8.02715	0.266244
$910$	0	0
$911$	11.5254	0.381854	0.190927	−	0.981604i	$-0.438851\pi$
$911$	11.5254	0.381854	0.190927	+	0.981604i	$0.438851\pi$
$912$	0.615957	0.0203964
$913$	−17.3840	−0.575328
$914$	16.0737	0.531670
$915$	0	0
$916$	23.3991	0.773128
$917$	−3.82371	−0.126270
$918$	35.9952	1.18802
$919$	8.28275	0.273223	0.136611	−	0.990625i	$-0.456379\pi$
$919$	8.28275	0.273223	0.136611	+	0.990625i	$0.456379\pi$
$920$	0	0
$921$	−8.66487	−0.285517
$922$	−27.5308	−0.906678
$923$	0	0
$924$	−0.274127	−0.00901811
$925$	0	0
$926$	−10.6595	−0.350293
$927$	−6.34913	−0.208533
$928$	−0.356896	−0.0117157
$929$	−25.3846	−0.832843	−0.416421	−	0.909172i	$-0.636716\pi$
$929$	−25.3846	−0.832843	−0.416421	+	0.909172i	$0.636716\pi$
$930$	0	0
$931$	−3.43834	−0.112687
$932$	−14.3913	−0.471404
$933$	−30.3779	−0.994529
$934$	10.7976	0.353309
$935$	0	0
$936$	0	0
$937$	21.4819	0.701782	0.350891	−	0.936416i	$-0.385879\pi$
$937$	21.4819	0.701782	0.350891	+	0.936416i	$0.385879\pi$
$938$	−0.767021	−0.0250441
$939$	−3.60388	−0.117608
$940$	0	0
$941$	−12.1148	−0.394932	−0.197466	−	0.980310i	$-0.563271\pi$
$941$	−12.1148	−0.394932	−0.197466	+	0.980310i	$0.563271\pi$
$942$	−21.7017	−0.707080
$943$	−60.6355	−1.97456
$944$	−4.61596	−0.150237
$945$	0	0
$946$	6.27413	0.203989
$947$	−23.4553	−0.762196	−0.381098	−	0.924535i	$-0.624454\pi$
$947$	−23.4553	−0.762196	−0.381098	+	0.924535i	$0.624454\pi$
$948$	3.65817	0.118812
$949$	0	0
$950$	0	0
$951$	6.24996	0.202669
$952$	−1.28621	−0.0416862
$953$	42.5086	1.37699	0.688494	−	0.725242i	$-0.258272\pi$
$953$	42.5086	1.37699	0.688494	+	0.725242i	$0.258272\pi$
$954$	14.3370	0.464179
$955$	0	0
$956$	−3.56033	−0.115149
$957$	0.493959	0.0159674
$958$	−9.08575	−0.293547
$959$	3.86938	0.124949
$960$	0	0
$961$	8.36467	0.269828
$962$	0	0
$963$	−25.9758	−0.837060
$964$	29.1890	0.940113
$965$	0	0
$966$	1.67025	0.0537395
$967$	−24.7827	−0.796957	−0.398479	−	0.917178i	$-0.630462\pi$
$967$	−24.7827	−0.796957	−0.398479	+	0.917178i	$0.630462\pi$
$968$	−9.76809	−0.313958
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−46.1124	−1.47982	−0.739909	−	0.672707i	$-0.765132\pi$
$971$	−46.1124	−1.47982	−0.739909	+	0.672707i	$0.765132\pi$
$972$	13.4155	0.430302
$973$	−3.64742	−0.116931
$974$	−40.8471	−1.30883
$975$	0	0
$976$	5.50365	0.176167
$977$	14.3827	0.460144	0.230072	−	0.973174i	$-0.426104\pi$
$977$	14.3827	0.460144	0.230072	+	0.973174i	$0.426104\pi$
$978$	−11.4330	−0.365586
$979$	−18.9578	−0.605894
$980$	0	0
$981$	9.73317	0.310756
$982$	29.0944	0.928439
$983$	−9.73615	−0.310535	−0.155268	−	0.987872i	$-0.549624\pi$
$983$	−9.73615	−0.310535	−0.155268	+	0.987872i	$0.549624\pi$
$984$	−11.1806	−0.356425
$985$	0	0
$986$	2.31767	0.0738096
$987$	−0.353912	−0.0112651
$988$	0	0
$989$	−38.2282	−1.21559
$990$	0	0
$991$	−25.7948	−0.819398	−0.409699	−	0.912221i	$-0.634366\pi$
$991$	−25.7948	−0.819398	−0.409699	+	0.912221i	$0.634366\pi$
$992$	−6.27413	−0.199204
$993$	29.9517	0.950488
$994$	−2.69441	−0.0854616
$995$	0	0
$996$	19.5308	0.618857
$997$	−29.8974	−0.946860	−0.473430	−	0.880832i	$-0.656984\pi$
$997$	−29.8974	−0.946860	−0.473430	+	0.880832i	$0.656984\pi$
$998$	−16.2150	−0.513278
$999$	−12.3043	−0.389289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bz.1.3		3
5.4	even	2		1690.2.a.q.1.1	✓	3
13.12	even	2		8450.2.a.bo.1.3		3
65.4	even	6		1690.2.e.o.991.3		6
65.9	even	6		1690.2.e.q.991.3		6
65.19	odd	12		1690.2.l.l.361.6		12
65.24	odd	12		1690.2.l.l.1161.6		12
65.29	even	6		1690.2.e.q.191.3		6
65.34	odd	4		1690.2.d.j.1351.1		6
65.44	odd	4		1690.2.d.j.1351.4		6
65.49	even	6		1690.2.e.o.191.3		6
65.54	odd	12		1690.2.l.l.1161.3		12
65.59	odd	12		1690.2.l.l.361.3		12
65.64	even	2		1690.2.a.s.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.q.1.1	✓	3	5.4	even	2
1690.2.a.s.1.1	yes	3	65.64	even	2
1690.2.d.j.1351.1		6	65.34	odd	4
1690.2.d.j.1351.4		6	65.44	odd	4
1690.2.e.o.191.3		6	65.49	even	6
1690.2.e.o.991.3		6	65.4	even	6
1690.2.e.q.191.3		6	65.29	even	6
1690.2.e.q.991.3		6	65.9	even	6
1690.2.l.l.361.3		12	65.59	odd	12
1690.2.l.l.361.6		12	65.19	odd	12
1690.2.l.l.1161.3		12	65.54	odd	12
1690.2.l.l.1161.6		12	65.24	odd	12
8450.2.a.bo.1.3		3	13.12	even	2
8450.2.a.bz.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.24698 q^{6} +0.198062 q^{7} +1.00000 q^{8} -1.44504 q^{9} -1.10992 q^{11} +1.24698 q^{12} +0.198062 q^{14} +1.00000 q^{16} -6.49396 q^{17} -1.44504 q^{18} +0.493959 q^{19} +0.246980 q^{21} -1.10992 q^{22} +6.76271 q^{23} +1.24698 q^{24} -5.54288 q^{27} +0.198062 q^{28} -0.356896 q^{29} -6.27413 q^{31} +1.00000 q^{32} -1.38404 q^{33} -6.49396 q^{34} -1.44504 q^{36} +2.21983 q^{37} +0.493959 q^{38} -8.96615 q^{41} +0.246980 q^{42} -5.65279 q^{43} -1.10992 q^{44} +6.76271 q^{46} -1.43296 q^{47} +1.24698 q^{48} -6.96077 q^{49} -8.09783 q^{51} -9.92154 q^{53} -5.54288 q^{54} +0.198062 q^{56} +0.615957 q^{57} -0.356896 q^{58} -4.61596 q^{59} +5.50365 q^{61} -6.27413 q^{62} -0.286208 q^{63} +1.00000 q^{64} -1.38404 q^{66} -3.87263 q^{67} -6.49396 q^{68} +8.43296 q^{69} -13.6039 q^{71} -1.44504 q^{72} +0.591794 q^{73} +2.21983 q^{74} +0.493959 q^{76} -0.219833 q^{77} +2.93362 q^{79} -2.57673 q^{81} -8.96615 q^{82} +15.6625 q^{83} +0.246980 q^{84} -5.65279 q^{86} -0.445042 q^{87} -1.10992 q^{88} +17.0804 q^{89} +6.76271 q^{92} -7.82371 q^{93} -1.43296 q^{94} +1.24698 q^{96} +13.5797 q^{97} -6.96077 q^{98} +1.60388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 5 q^{7} + 3 q^{8} - 4 q^{9} - 4 q^{11} - q^{12} + 5 q^{14} + 3 q^{16} - 10 q^{17} - 4 q^{18} - 8 q^{19} - 4 q^{21} - 4 q^{22} + 3 q^{23} - q^{24} + 2 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 5 * q^7 + 3 * q^8 - 4 * q^9 - 4 * q^11 - q^12 + 5 * q^14 + 3 * q^16 - 10 * q^17 - 4 * q^18 - 8 * q^19 - 4 * q^21 - 4 * q^22 + 3 * q^23 - q^24 + 2 * q^27 + 5 * q^28 + 3 * q^29 - 8 * q^31 + 3 * q^32 + 6 * q^33 - 10 * q^34 - 4 * q^36 + 8 * q^37 - 8 * q^38 - 11 * q^41 - 4 * q^42 + q^43 - 4 * q^44 + 3 * q^46 + 15 * q^47 - q^48 - 8 * q^49 - 6 * q^51 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 12 * q^57 + 3 * q^58 - 24 * q^59 - 15 * q^61 - 8 * q^62 - 9 * q^63 + 3 * q^64 + 6 * q^66 + 5 * q^67 - 10 * q^68 + 6 * q^69 - 32 * q^71 - 4 * q^72 - 26 * q^73 + 8 * q^74 - 8 * q^76 - 2 * q^77 + 2 * q^79 - 5 * q^81 - 11 * q^82 + 7 * q^83 - 4 * q^84 + q^86 - q^87 - 4 * q^88 + 17 * q^89 + 3 * q^92 - 16 * q^93 + 15 * q^94 - q^96 - 6 * q^97 - 8 * q^98 - 4 * q^99

Introduction

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