LMFDB - Embedded newform 8450.2.a.da.1.2

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 = [9,9,7,9,0,7,1,9,8,0,4,7,0,1,0,9,12] 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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 $ x^9 - x^8 - 14x^7 + 17x^6 + 53x^5 - 69x^4 - 33x^3 + 26x^2 + 8*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.2
Root			$2.08395$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.49535 q^{3} +1.00000 q^{4} -1.49535 q^{6} +1.01867 q^{7} +1.00000 q^{8} -0.763924 q^{9} +2.42025 q^{11} -1.49535 q^{12} +1.01867 q^{14} +1.00000 q^{16} +5.04699 q^{17} -0.763924 q^{18} +7.36506 q^{19} -1.52327 q^{21} +2.42025 q^{22} +4.52359 q^{23} -1.49535 q^{24} +5.62839 q^{27} +1.01867 q^{28} +6.38898 q^{29} -6.28274 q^{31} +1.00000 q^{32} -3.61912 q^{33} +5.04699 q^{34} -0.763924 q^{36} +1.80149 q^{37} +7.36506 q^{38} -0.399553 q^{41} -1.52327 q^{42} -8.41814 q^{43} +2.42025 q^{44} +4.52359 q^{46} -0.146040 q^{47} -1.49535 q^{48} -5.96231 q^{49} -7.54702 q^{51} +1.52601 q^{53} +5.62839 q^{54} +1.01867 q^{56} -11.0134 q^{57} +6.38898 q^{58} +3.23605 q^{59} -14.1612 q^{61} -6.28274 q^{62} -0.778187 q^{63} +1.00000 q^{64} -3.61912 q^{66} -6.59001 q^{67} +5.04699 q^{68} -6.76435 q^{69} +1.36946 q^{71} -0.763924 q^{72} +15.9009 q^{73} +1.80149 q^{74} +7.36506 q^{76} +2.46544 q^{77} -3.53802 q^{79} -6.12465 q^{81} -0.399553 q^{82} -4.64985 q^{83} -1.52327 q^{84} -8.41814 q^{86} -9.55376 q^{87} +2.42025 q^{88} +5.60882 q^{89} +4.52359 q^{92} +9.39490 q^{93} -0.146040 q^{94} -1.49535 q^{96} -8.27379 q^{97} -5.96231 q^{98} -1.84889 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + 4 q^{11} + 7 q^{12} + q^{14} + 9 q^{16} + 12 q^{17} + 8 q^{18} + 6 q^{19} + 8 q^{21} + 4 q^{22} + 11 q^{23} + 7 q^{24} + 34 q^{27}+ \cdots + 16 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + 7 * q^3 + 9 * q^4 + 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 + 4 * q^11 + 7 * q^12 + q^14 + 9 * q^16 + 12 * q^17 + 8 * q^18 + 6 * q^19 + 8 * q^21 + 4 * q^22 + 11 * q^23 + 7 * q^24 + 34 * q^27 + q^28 + 15 * q^29 - 6 * q^31 + 9 * q^32 - 14 * q^33 + 12 * q^34 + 8 * q^36 - 10 * q^37 + 6 * q^38 + 27 * q^41 + 8 * q^42 + 41 * q^43 + 4 * q^44 + 11 * q^46 + 5 * q^47 + 7 * q^48 - 10 * q^49 + 12 * q^51 + 18 * q^53 + 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 + 20 * q^59 - 19 * q^61 - 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 + 12 * q^68 - 14 * q^69 - 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 + 6 * q^76 + 30 * q^77 - 14 * q^79 + 53 * q^81 + 27 * q^82 + q^83 + 8 * q^84 + 41 * q^86 + 33 * q^87 + 4 * q^88 + 43 * q^89 + 11 * q^92 - 28 * q^93 + 5 * q^94 + 7 * q^96 + 6 * q^97 - 10 * q^98 + 16 * 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
$3$	−1.49535	−0.863342	−0.431671	−	0.902031i	$-0.642076\pi$
$3$	−1.49535	−0.863342	−0.431671	+	0.902031i	$0.642076\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.49535	−0.610475
$7$	1.01867	0.385021	0.192511	−	0.981295i	$-0.438337\pi$
$7$	1.01867	0.385021	0.192511	+	0.981295i	$0.438337\pi$
$8$	1.00000	0.353553
$9$	−0.763924	−0.254641
$10$	0	0
$11$	2.42025	0.729733	0.364866	−	0.931060i	$-0.381115\pi$
$11$	2.42025	0.729733	0.364866	+	0.931060i	$0.381115\pi$
$12$	−1.49535	−0.431671
$13$	0	0
$13$	0	0
$14$	1.01867	0.272251
$15$	0	0
$16$	1.00000	0.250000
$17$	5.04699	1.22407	0.612037	−	0.790829i	$-0.290350\pi$
$17$	5.04699	1.22407	0.612037	+	0.790829i	$0.290350\pi$
$18$	−0.763924	−0.180059
$19$	7.36506	1.68966	0.844830	−	0.535034i	$-0.179701\pi$
$19$	7.36506	1.68966	0.844830	+	0.535034i	$0.179701\pi$
$20$	0	0
$21$	−1.52327	−0.332405
$22$	2.42025	0.515999
$23$	4.52359	0.943233	0.471616	−	0.881804i	$-0.343671\pi$
$23$	4.52359	0.943233	0.471616	+	0.881804i	$0.343671\pi$
$24$	−1.49535	−0.305237
$25$	0	0
$26$	0	0
$27$	5.62839	1.08318
$28$	1.01867	0.192511
$29$	6.38898	1.18640	0.593202	−	0.805054i	$-0.297864\pi$
$29$	6.38898	1.18640	0.593202	+	0.805054i	$0.297864\pi$
$30$	0	0
$31$	−6.28274	−1.12841	−0.564207	−	0.825634i	$-0.690818\pi$
$31$	−6.28274	−1.12841	−0.564207	+	0.825634i	$0.690818\pi$
$32$	1.00000	0.176777
$33$	−3.61912	−0.630009
$34$	5.04699	0.865551
$35$	0	0
$36$	−0.763924	−0.127321
$37$	1.80149	0.296163	0.148081	−	0.988975i	$-0.452690\pi$
$37$	1.80149	0.296163	0.148081	+	0.988975i	$0.452690\pi$
$38$	7.36506	1.19477
$39$	0	0
$40$	0	0
$41$	−0.399553	−0.0623997	−0.0311999	−	0.999513i	$-0.509933\pi$
$41$	−0.399553	−0.0623997	−0.0311999	+	0.999513i	$0.509933\pi$
$42$	−1.52327	−0.235046
$43$	−8.41814	−1.28375	−0.641877	−	0.766808i	$-0.721844\pi$
$43$	−8.41814	−1.28375	−0.641877	+	0.766808i	$0.721844\pi$
$44$	2.42025	0.364866
$45$	0	0
$46$	4.52359	0.666966
$47$	−0.146040	−0.0213021	−0.0106511	−	0.999943i	$-0.503390\pi$
$47$	−0.146040	−0.0213021	−0.0106511	+	0.999943i	$0.503390\pi$
$48$	−1.49535	−0.215835
$49$	−5.96231	−0.851758
$50$	0	0
$51$	−7.54702	−1.05679
$52$	0	0
$53$	1.52601	0.209614	0.104807	−	0.994493i	$-0.466578\pi$
$53$	1.52601	0.209614	0.104807	+	0.994493i	$0.466578\pi$
$54$	5.62839	0.765927
$55$	0	0
$56$	1.01867	0.136126
$57$	−11.0134	−1.45875
$58$	6.38898	0.838914
$59$	3.23605	0.421298	0.210649	−	0.977562i	$-0.432442\pi$
$59$	3.23605	0.421298	0.210649	+	0.977562i	$0.432442\pi$
$60$	0	0
$61$	−14.1612	−1.81316	−0.906580	−	0.422034i	$-0.861316\pi$
$61$	−14.1612	−1.81316	−0.906580	+	0.422034i	$0.861316\pi$
$62$	−6.28274	−0.797909
$63$	−0.778187	−0.0980424
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.61912	−0.445483
$67$	−6.59001	−0.805098	−0.402549	−	0.915398i	$-0.631876\pi$
$67$	−6.59001	−0.805098	−0.402549	+	0.915398i	$0.631876\pi$
$68$	5.04699	0.612037
$69$	−6.76435	−0.814332
$70$	0	0
$71$	1.36946	0.162525	0.0812623	−	0.996693i	$-0.474105\pi$
$71$	1.36946	0.162525	0.0812623	+	0.996693i	$0.474105\pi$
$72$	−0.763924	−0.0900293
$73$	15.9009	1.86106	0.930531	−	0.366213i	$-0.119346\pi$
$73$	15.9009	1.86106	0.930531	+	0.366213i	$0.119346\pi$
$74$	1.80149	0.209419
$75$	0	0
$76$	7.36506	0.844830
$77$	2.46544	0.280963
$78$	0	0
$79$	−3.53802	−0.398059	−0.199029	−	0.979994i	$-0.563779\pi$
$79$	−3.53802	−0.398059	−0.199029	+	0.979994i	$0.563779\pi$
$80$	0	0
$81$	−6.12465	−0.680516
$82$	−0.399553	−0.0441233
$83$	−4.64985	−0.510387	−0.255194	−	0.966890i	$-0.582139\pi$
$83$	−4.64985	−0.510387	−0.255194	+	0.966890i	$0.582139\pi$
$84$	−1.52327	−0.166203
$85$	0	0
$86$	−8.41814	−0.907751
$87$	−9.55376	−1.02427
$88$	2.42025	0.258000
$89$	5.60882	0.594533	0.297267	−	0.954794i	$-0.403925\pi$
$89$	5.60882	0.594533	0.297267	+	0.954794i	$0.403925\pi$
$90$	0	0
$91$	0	0
$92$	4.52359	0.471616
$93$	9.39490	0.974206
$94$	−0.146040	−0.0150629
$95$	0	0
$96$	−1.49535	−0.152619
$97$	−8.27379	−0.840076	−0.420038	−	0.907507i	$-0.637983\pi$
$97$	−8.27379	−0.840076	−0.420038	+	0.907507i	$0.637983\pi$
$98$	−5.96231	−0.602284
$99$	−1.84889	−0.185820
$100$	0	0
$101$	−19.9369	−1.98379	−0.991896	−	0.127053i	$-0.959448\pi$
$101$	−19.9369	−1.98379	−0.991896	+	0.127053i	$0.959448\pi$
$102$	−7.54702	−0.747266
$103$	14.4702	1.42579	0.712893	−	0.701272i	$-0.247384\pi$
$103$	14.4702	1.42579	0.712893	+	0.701272i	$0.247384\pi$
$104$	0	0
$105$	0	0
$106$	1.52601	0.148219
$107$	6.49059	0.627469	0.313734	−	0.949511i	$-0.398420\pi$
$107$	6.49059	0.627469	0.313734	+	0.949511i	$0.398420\pi$
$108$	5.62839	0.541592
$109$	13.8750	1.32899	0.664494	−	0.747293i	$-0.268647\pi$
$109$	13.8750	1.32899	0.664494	+	0.747293i	$0.268647\pi$
$110$	0	0
$111$	−2.69386	−0.255690
$112$	1.01867	0.0962554
$113$	−12.0565	−1.13418	−0.567089	−	0.823656i	$-0.691931\pi$
$113$	−12.0565	−1.13418	−0.567089	+	0.823656i	$0.691931\pi$
$114$	−11.0134	−1.03149
$115$	0	0
$116$	6.38898	0.593202
$117$	0	0
$118$	3.23605	0.297903
$119$	5.14122	0.471295
$120$	0	0
$121$	−5.14239	−0.467490
$122$	−14.1612	−1.28210
$123$	0.597472	0.0538723
$124$	−6.28274	−0.564207
$125$	0	0
$126$	−0.778187	−0.0693264
$127$	18.2962	1.62352	0.811761	−	0.583990i	$-0.198509\pi$
$127$	18.2962	1.62352	0.811761	+	0.583990i	$0.198509\pi$
$128$	1.00000	0.0883883
$129$	12.5881	1.10832
$130$	0	0
$131$	21.8260	1.90694	0.953472	−	0.301481i	$-0.0974808\pi$
$131$	21.8260	1.90694	0.953472	+	0.301481i	$0.0974808\pi$
$132$	−3.61912	−0.315004
$133$	7.50257	0.650556
$134$	−6.59001	−0.569290
$135$	0	0
$136$	5.04699	0.432776
$137$	8.05446	0.688139	0.344070	−	0.938944i	$-0.388194\pi$
$137$	8.05446	0.688139	0.344070	+	0.938944i	$0.388194\pi$
$138$	−6.76435	−0.575820
$139$	−12.8464	−1.08962	−0.544809	−	0.838560i	$-0.683398\pi$
$139$	−12.8464	−1.08962	−0.544809	+	0.838560i	$0.683398\pi$
$140$	0	0
$141$	0.218381	0.0183910
$142$	1.36946	0.114922
$143$	0	0
$144$	−0.763924	−0.0636603
$145$	0	0
$146$	15.9009	1.31597
$147$	8.91575	0.735358
$148$	1.80149	0.148081
$149$	11.6532	0.954669	0.477335	−	0.878722i	$-0.341603\pi$
$149$	11.6532	0.954669	0.477335	+	0.878722i	$0.341603\pi$
$150$	0	0
$151$	−3.93672	−0.320365	−0.160183	−	0.987087i	$-0.551208\pi$
$151$	−3.93672	−0.320365	−0.160183	+	0.987087i	$0.551208\pi$
$152$	7.36506	0.597385
$153$	−3.85552	−0.311700
$154$	2.46544	0.198671
$155$	0	0
$156$	0	0
$157$	1.86131	0.148548	0.0742742	−	0.997238i	$-0.476336\pi$
$157$	1.86131	0.148548	0.0742742	+	0.997238i	$0.476336\pi$
$158$	−3.53802	−0.281470
$159$	−2.28193	−0.180968
$160$	0	0
$161$	4.60805	0.363165
$162$	−6.12465	−0.481198
$163$	−23.0167	−1.80280	−0.901402	−	0.432984i	$-0.857461\pi$
$163$	−23.0167	−1.80280	−0.901402	+	0.432984i	$0.857461\pi$
$164$	−0.399553	−0.0311999
$165$	0	0
$166$	−4.64985	−0.360898
$167$	6.11064	0.472856	0.236428	−	0.971649i	$-0.424023\pi$
$167$	6.11064	0.472856	0.236428	+	0.971649i	$0.424023\pi$
$168$	−1.52327	−0.117523
$169$	0	0
$170$	0	0
$171$	−5.62635	−0.430257
$172$	−8.41814	−0.641877
$173$	23.0755	1.75440	0.877200	−	0.480125i	$-0.159409\pi$
$173$	23.0755	1.75440	0.877200	+	0.480125i	$0.159409\pi$
$174$	−9.55376	−0.724269
$175$	0	0
$176$	2.42025	0.182433
$177$	−4.83904	−0.363724
$178$	5.60882	0.420399
$179$	7.81690	0.584263	0.292131	−	0.956378i	$-0.405636\pi$
$179$	7.81690	0.584263	0.292131	+	0.956378i	$0.405636\pi$
$180$	0	0
$181$	−5.21467	−0.387603	−0.193802	−	0.981041i	$-0.562082\pi$
$181$	−5.21467	−0.387603	−0.193802	+	0.981041i	$0.562082\pi$
$182$	0	0
$183$	21.1760	1.56538
$184$	4.52359	0.333483
$185$	0	0
$186$	9.39490	0.688868
$187$	12.2150	0.893248
$188$	−0.146040	−0.0106511
$189$	5.73348	0.417049
$190$	0	0
$191$	10.4143	0.753554	0.376777	−	0.926304i	$-0.377032\pi$
$191$	10.4143	0.753554	0.376777	+	0.926304i	$0.377032\pi$
$192$	−1.49535	−0.107918
$193$	1.40263	0.100964	0.0504819	−	0.998725i	$-0.483924\pi$
$193$	1.40263	0.100964	0.0504819	+	0.998725i	$0.483924\pi$
$194$	−8.27379	−0.594023
$195$	0	0
$196$	−5.96231	−0.425879
$197$	25.3455	1.80579	0.902897	−	0.429857i	$-0.141436\pi$
$197$	25.3455	1.80579	0.902897	+	0.429857i	$0.141436\pi$
$198$	−1.84889	−0.131395
$199$	13.7630	0.975636	0.487818	−	0.872945i	$-0.337793\pi$
$199$	13.7630	0.975636	0.487818	+	0.872945i	$0.337793\pi$
$200$	0	0
$201$	9.85438	0.695074
$202$	−19.9369	−1.40275
$203$	6.50826	0.456791
$204$	−7.54702	−0.528397
$205$	0	0
$206$	14.4702	1.00818
$207$	−3.45568	−0.240186
$208$	0	0
$209$	17.8253	1.23300
$210$	0	0
$211$	9.82347	0.676275	0.338138	−	0.941097i	$-0.390203\pi$
$211$	9.82347	0.676275	0.338138	+	0.941097i	$0.390203\pi$
$212$	1.52601	0.104807
$213$	−2.04782	−0.140314
$214$	6.49059	0.443688
$215$	0	0
$216$	5.62839	0.382963
$217$	−6.40004	−0.434463
$218$	13.8750	0.939737
$219$	−23.7775	−1.60673
$220$	0	0
$221$	0	0
$222$	−2.69386	−0.180800
$223$	19.3560	1.29618	0.648088	−	0.761565i	$-0.275569\pi$
$223$	19.3560	1.29618	0.648088	+	0.761565i	$0.275569\pi$
$224$	1.01867	0.0680628
$225$	0	0
$226$	−12.0565	−0.801986
$227$	20.7535	1.37746	0.688729	−	0.725019i	$-0.258169\pi$
$227$	20.7535	1.37746	0.688729	+	0.725019i	$0.258169\pi$
$228$	−11.0134	−0.729377
$229$	−23.2933	−1.53927	−0.769634	−	0.638486i	$-0.779561\pi$
$229$	−23.2933	−1.53927	−0.769634	+	0.638486i	$0.779561\pi$
$230$	0	0
$231$	−3.68670	−0.242567
$232$	6.38898	0.419457
$233$	0.585969	0.0383881	0.0191941	−	0.999816i	$-0.493890\pi$
$233$	0.585969	0.0383881	0.0191941	+	0.999816i	$0.493890\pi$
$234$	0	0
$235$	0	0
$236$	3.23605	0.210649
$237$	5.29059	0.343660
$238$	5.14122	0.333256
$239$	−9.56019	−0.618397	−0.309199	−	0.950998i	$-0.600061\pi$
$239$	−9.56019	−0.618397	−0.309199	+	0.950998i	$0.600061\pi$
$240$	0	0
$241$	−11.5135	−0.741650	−0.370825	−	0.928703i	$-0.620925\pi$
$241$	−11.5135	−0.741650	−0.370825	+	0.928703i	$0.620925\pi$
$242$	−5.14239	−0.330565
$243$	−7.72667	−0.495666
$244$	−14.1612	−0.906580
$245$	0	0
$246$	0.597472	0.0380935
$247$	0	0
$248$	−6.28274	−0.398954
$249$	6.95316	0.440639
$250$	0	0
$251$	−15.0024	−0.946942	−0.473471	−	0.880809i	$-0.656999\pi$
$251$	−15.0024	−0.946942	−0.473471	+	0.880809i	$0.656999\pi$
$252$	−0.778187	−0.0490212
$253$	10.9482	0.688308
$254$	18.2962	1.14800
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.902680	−0.0563077	−0.0281538	−	0.999604i	$-0.508963\pi$
$257$	−0.902680	−0.0563077	−0.0281538	+	0.999604i	$0.508963\pi$
$258$	12.5881	0.783699
$259$	1.83512	0.114029
$260$	0	0
$261$	−4.88069	−0.302107
$262$	21.8260	1.34841
$263$	−17.5243	−1.08059	−0.540297	−	0.841475i	$-0.681688\pi$
$263$	−17.5243	−1.08059	−0.540297	+	0.841475i	$0.681688\pi$
$264$	−3.61912	−0.222742
$265$	0	0
$266$	7.50257	0.460012
$267$	−8.38715	−0.513285
$268$	−6.59001	−0.402549
$269$	3.17990	0.193882	0.0969410	−	0.995290i	$-0.469094\pi$
$269$	3.17990	0.193882	0.0969410	+	0.995290i	$0.469094\pi$
$270$	0	0
$271$	−13.8015	−0.838382	−0.419191	−	0.907898i	$-0.637686\pi$
$271$	−13.8015	−0.838382	−0.419191	+	0.907898i	$0.637686\pi$
$272$	5.04699	0.306019
$273$	0	0
$274$	8.05446	0.486588
$275$	0	0
$276$	−6.76435	−0.407166
$277$	−4.77040	−0.286626	−0.143313	−	0.989677i	$-0.545775\pi$
$277$	−4.77040	−0.286626	−0.143313	+	0.989677i	$0.545775\pi$
$278$	−12.8464	−0.770476
$279$	4.79954	0.287341
$280$	0	0
$281$	−5.70927	−0.340586	−0.170293	−	0.985393i	$-0.554471\pi$
$281$	−5.70927	−0.340586	−0.170293	+	0.985393i	$0.554471\pi$
$282$	0.218381	0.0130044
$283$	21.2228	1.26157	0.630783	−	0.775960i	$-0.282734\pi$
$283$	21.2228	1.26157	0.630783	+	0.775960i	$0.282734\pi$
$284$	1.36946	0.0812623
$285$	0	0
$286$	0	0
$287$	−0.407013	−0.0240252
$288$	−0.763924	−0.0450147
$289$	8.47209	0.498359
$290$	0	0
$291$	12.3722	0.725272
$292$	15.9009	0.930531
$293$	−23.9835	−1.40113	−0.700565	−	0.713588i	$-0.747069\pi$
$293$	−23.9835	−1.40113	−0.700565	+	0.713588i	$0.747069\pi$
$294$	8.91575	0.519977
$295$	0	0
$296$	1.80149	0.104709
$297$	13.6221	0.790435
$298$	11.6532	0.675053
$299$	0	0
$300$	0	0
$301$	−8.57531	−0.494273
$302$	−3.93672	−0.226533
$303$	29.8126	1.71269
$304$	7.36506	0.422415
$305$	0	0
$306$	−3.85552	−0.220405
$307$	17.0892	0.975334	0.487667	−	0.873030i	$-0.337848\pi$
$307$	17.0892	0.975334	0.487667	+	0.873030i	$0.337848\pi$
$308$	2.46544	0.140481
$309$	−21.6380	−1.23094
$310$	0	0
$311$	5.17968	0.293713	0.146856	−	0.989158i	$-0.453084\pi$
$311$	5.17968	0.293713	0.146856	+	0.989158i	$0.453084\pi$
$312$	0	0
$313$	33.7809	1.90941	0.954704	−	0.297556i	$-0.0961717\pi$
$313$	33.7809	1.90941	0.954704	+	0.297556i	$0.0961717\pi$
$314$	1.86131	0.105040
$315$	0	0
$316$	−3.53802	−0.199029
$317$	23.3968	1.31409	0.657047	−	0.753850i	$-0.271805\pi$
$317$	23.3968	1.31409	0.657047	+	0.753850i	$0.271805\pi$
$318$	−2.28193	−0.127964
$319$	15.4629	0.865757
$320$	0	0
$321$	−9.70571	−0.541720
$322$	4.60805	0.256796
$323$	37.1714	2.06827
$324$	−6.12465	−0.340258
$325$	0	0
$326$	−23.0167	−1.27477
$327$	−20.7481	−1.14737
$328$	−0.399553	−0.0220616
$329$	−0.148767	−0.00820178
$330$	0	0
$331$	−5.46645	−0.300463	−0.150232	−	0.988651i	$-0.548002\pi$
$331$	−5.46645	−0.300463	−0.150232	+	0.988651i	$0.548002\pi$
$332$	−4.64985	−0.255194
$333$	−1.37620	−0.0754153
$334$	6.11064	0.334359
$335$	0	0
$336$	−1.52327	−0.0831013
$337$	25.4138	1.38438	0.692188	−	0.721717i	$-0.256647\pi$
$337$	25.4138	1.38438	0.692188	+	0.721717i	$0.256647\pi$
$338$	0	0
$339$	18.0287	0.979184
$340$	0	0
$341$	−15.2058	−0.823440
$342$	−5.62635	−0.304238
$343$	−13.2043	−0.712967
$344$	−8.41814	−0.453876
$345$	0	0
$346$	23.0755	1.24055
$347$	−17.1079	−0.918402	−0.459201	−	0.888332i	$-0.651864\pi$
$347$	−17.1079	−0.918402	−0.459201	+	0.888332i	$0.651864\pi$
$348$	−9.55376	−0.512136
$349$	0.935233	0.0500619	0.0250309	−	0.999687i	$-0.492032\pi$
$349$	0.935233	0.0500619	0.0250309	+	0.999687i	$0.492032\pi$
$350$	0	0
$351$	0	0
$352$	2.42025	0.129000
$353$	−0.243550	−0.0129628	−0.00648142	−	0.999979i	$-0.502063\pi$
$353$	−0.243550	−0.0129628	−0.00648142	+	0.999979i	$0.502063\pi$
$354$	−4.83904	−0.257192
$355$	0	0
$356$	5.60882	0.297267
$357$	−7.68793	−0.406889
$358$	7.81690	0.413136
$359$	−14.7698	−0.779520	−0.389760	−	0.920916i	$-0.627442\pi$
$359$	−14.7698	−0.779520	−0.389760	+	0.920916i	$0.627442\pi$
$360$	0	0
$361$	35.2441	1.85495
$362$	−5.21467	−0.274077
$363$	7.68968	0.403603
$364$	0	0
$365$	0	0
$366$	21.1760	1.10689
$367$	−3.29260	−0.171872	−0.0859362	−	0.996301i	$-0.527388\pi$
$367$	−3.29260	−0.171872	−0.0859362	+	0.996301i	$0.527388\pi$
$368$	4.52359	0.235808
$369$	0.305228	0.0158896
$370$	0	0
$371$	1.55450	0.0807059
$372$	9.39490	0.487103
$373$	−18.9067	−0.978950	−0.489475	−	0.872017i	$-0.662812\pi$
$373$	−18.9067	−0.978950	−0.489475	+	0.872017i	$0.662812\pi$
$374$	12.2150	0.631621
$375$	0	0
$376$	−0.146040	−0.00753144
$377$	0	0
$378$	5.73348	0.294898
$379$	−13.7733	−0.707489	−0.353744	−	0.935342i	$-0.615092\pi$
$379$	−13.7733	−0.707489	−0.353744	+	0.935342i	$0.615092\pi$
$380$	0	0
$381$	−27.3592	−1.40165
$382$	10.4143	0.532843
$383$	−33.3243	−1.70279	−0.851395	−	0.524525i	$-0.824243\pi$
$383$	−33.3243	−1.70279	−0.851395	+	0.524525i	$0.824243\pi$
$384$	−1.49535	−0.0763093
$385$	0	0
$386$	1.40263	0.0713922
$387$	6.43082	0.326897
$388$	−8.27379	−0.420038
$389$	−1.91117	−0.0969004	−0.0484502	−	0.998826i	$-0.515428\pi$
$389$	−1.91117	−0.0969004	−0.0484502	+	0.998826i	$0.515428\pi$
$390$	0	0
$391$	22.8305	1.15459
$392$	−5.96231	−0.301142
$393$	−32.6375	−1.64634
$394$	25.3455	1.27689
$395$	0	0
$396$	−1.84889	−0.0929101
$397$	−13.5405	−0.679579	−0.339789	−	0.940502i	$-0.610356\pi$
$397$	−13.5405	−0.679579	−0.339789	+	0.940502i	$0.610356\pi$
$398$	13.7630	0.689878
$399$	−11.2190	−0.561652
$400$	0	0
$401$	−28.9798	−1.44718	−0.723592	−	0.690228i	$-0.757510\pi$
$401$	−28.9798	−1.44718	−0.723592	+	0.690228i	$0.757510\pi$
$402$	9.85438	0.491492
$403$	0	0
$404$	−19.9369	−0.991896
$405$	0	0
$406$	6.50826	0.323000
$407$	4.36005	0.216120
$408$	−7.54702	−0.373633
$409$	−9.77252	−0.483220	−0.241610	−	0.970373i	$-0.577675\pi$
$409$	−9.77252	−0.483220	−0.241610	+	0.970373i	$0.577675\pi$
$410$	0	0
$411$	−12.0442	−0.594099
$412$	14.4702	0.712893
$413$	3.29647	0.162209
$414$	−3.45568	−0.169837
$415$	0	0
$416$	0	0
$417$	19.2099	0.940712
$418$	17.8253	0.871863
$419$	−24.9288	−1.21785	−0.608926	−	0.793227i	$-0.708399\pi$
$419$	−24.9288	−1.21785	−0.608926	+	0.793227i	$0.708399\pi$
$420$	0	0
$421$	12.7436	0.621087	0.310544	−	0.950559i	$-0.399489\pi$
$421$	12.7436	0.621087	0.310544	+	0.950559i	$0.399489\pi$
$422$	9.82347	0.478199
$423$	0.111564	0.00542441
$424$	1.52601	0.0741097
$425$	0	0
$426$	−2.04782	−0.0992172
$427$	−14.4256	−0.698105
$428$	6.49059	0.313734
$429$	0	0
$430$	0	0
$431$	22.2564	1.07205	0.536026	−	0.844202i	$-0.319925\pi$
$431$	22.2564	1.07205	0.536026	+	0.844202i	$0.319925\pi$
$432$	5.62839	0.270796
$433$	30.4130	1.46156	0.730778	−	0.682615i	$-0.239157\pi$
$433$	30.4130	1.46156	0.730778	+	0.682615i	$0.239157\pi$
$434$	−6.40004	−0.307212
$435$	0	0
$436$	13.8750	0.664494
$437$	33.3165	1.59374
$438$	−23.7775	−1.13613
$439$	−12.5984	−0.601287	−0.300644	−	0.953737i	$-0.597201\pi$
$439$	−12.5984	−0.601287	−0.300644	+	0.953737i	$0.597201\pi$
$440$	0	0
$441$	4.55475	0.216893
$442$	0	0
$443$	2.42243	0.115093	0.0575467	−	0.998343i	$-0.481672\pi$
$443$	2.42243	0.115093	0.0575467	+	0.998343i	$0.481672\pi$
$444$	−2.69386	−0.127845
$445$	0	0
$446$	19.3560	0.916535
$447$	−17.4257	−0.824206
$448$	1.01867	0.0481277
$449$	−29.0963	−1.37314	−0.686568	−	0.727065i	$-0.740884\pi$
$449$	−29.0963	−1.37314	−0.686568	+	0.727065i	$0.740884\pi$
$450$	0	0
$451$	−0.967019	−0.0455351
$452$	−12.0565	−0.567089
$453$	5.88677	0.276585
$454$	20.7535	0.974010
$455$	0	0
$456$	−11.0134	−0.515747
$457$	−5.25707	−0.245916	−0.122958	−	0.992412i	$-0.539238\pi$
$457$	−5.25707	−0.245916	−0.122958	+	0.992412i	$0.539238\pi$
$458$	−23.2933	−1.08843
$459$	28.4064	1.32590
$460$	0	0
$461$	21.6948	1.01043	0.505213	−	0.862995i	$-0.331414\pi$
$461$	21.6948	1.01043	0.505213	+	0.862995i	$0.331414\pi$
$462$	−3.68670	−0.171521
$463$	−6.17734	−0.287085	−0.143543	−	0.989644i	$-0.545849\pi$
$463$	−6.17734	−0.287085	−0.143543	+	0.989644i	$0.545849\pi$
$464$	6.38898	0.296601
$465$	0	0
$466$	0.585969	0.0271445
$467$	9.01521	0.417174	0.208587	−	0.978004i	$-0.433114\pi$
$467$	9.01521	0.417174	0.208587	+	0.978004i	$0.433114\pi$
$468$	0	0
$469$	−6.71305	−0.309980
$470$	0	0
$471$	−2.78331	−0.128248
$472$	3.23605	0.148951
$473$	−20.3740	−0.936798
$474$	5.29059	0.243005
$475$	0	0
$476$	5.14122	0.235647
$477$	−1.16576	−0.0533764
$478$	−9.56019	−0.437273
$479$	14.3406	0.655240	0.327620	−	0.944810i	$-0.393753\pi$
$479$	14.3406	0.655240	0.327620	+	0.944810i	$0.393753\pi$
$480$	0	0
$481$	0	0
$482$	−11.5135	−0.524426
$483$	−6.89065	−0.313535
$484$	−5.14239	−0.233745
$485$	0	0
$486$	−7.72667	−0.350489
$487$	−15.5121	−0.702921	−0.351460	−	0.936203i	$-0.614315\pi$
$487$	−15.5121	−0.702921	−0.351460	+	0.936203i	$0.614315\pi$
$488$	−14.1612	−0.641049
$489$	34.4180	1.55644
$490$	0	0
$491$	2.60458	0.117543	0.0587715	−	0.998271i	$-0.481282\pi$
$491$	2.60458	0.117543	0.0587715	+	0.998271i	$0.481282\pi$
$492$	0.597472	0.0269361
$493$	32.2451	1.45225
$494$	0	0
$495$	0	0
$496$	−6.28274	−0.282103
$497$	1.39503	0.0625755
$498$	6.95316	0.311579
$499$	4.88458	0.218664	0.109332	−	0.994005i	$-0.465129\pi$
$499$	4.88458	0.218664	0.109332	+	0.994005i	$0.465129\pi$
$500$	0	0
$501$	−9.13756	−0.408236
$502$	−15.0024	−0.669589
$503$	−1.67768	−0.0748041	−0.0374020	−	0.999300i	$-0.511908\pi$
$503$	−1.67768	−0.0748041	−0.0374020	+	0.999300i	$0.511908\pi$
$504$	−0.778187	−0.0346632
$505$	0	0
$506$	10.9482	0.486707
$507$	0	0
$508$	18.2962	0.811761
$509$	−24.0762	−1.06716	−0.533580	−	0.845750i	$-0.679154\pi$
$509$	−24.0762	−1.06716	−0.533580	+	0.845750i	$0.679154\pi$
$510$	0	0
$511$	16.1978	0.716549
$512$	1.00000	0.0441942
$513$	41.4534	1.83021
$514$	−0.902680	−0.0398155
$515$	0	0
$516$	12.5881	0.554159
$517$	−0.353454	−0.0155449
$518$	1.83512	0.0806307
$519$	−34.5060	−1.51465
$520$	0	0
$521$	4.74906	0.208060	0.104030	−	0.994574i	$-0.466826\pi$
$521$	4.74906	0.208060	0.104030	+	0.994574i	$0.466826\pi$
$522$	−4.88069	−0.213622
$523$	21.1351	0.924174	0.462087	−	0.886835i	$-0.347101\pi$
$523$	21.1351	0.924174	0.462087	+	0.886835i	$0.347101\pi$
$524$	21.8260	0.953472
$525$	0	0
$526$	−17.5243	−0.764095
$527$	−31.7089	−1.38126
$528$	−3.61912	−0.157502
$529$	−2.53718	−0.110312
$530$	0	0
$531$	−2.47210	−0.107280
$532$	7.50257	0.325278
$533$	0	0
$534$	−8.38715	−0.362948
$535$	0	0
$536$	−6.59001	−0.284645
$537$	−11.6890	−0.504418
$538$	3.17990	0.137095
$539$	−14.4303	−0.621556
$540$	0	0
$541$	16.3708	0.703835	0.351917	−	0.936031i	$-0.385530\pi$
$541$	16.3708	0.703835	0.351917	+	0.936031i	$0.385530\pi$
$542$	−13.8015	−0.592825
$543$	7.79776	0.334634
$544$	5.04699	0.216388
$545$	0	0
$546$	0	0
$547$	−25.9050	−1.10762	−0.553810	−	0.832643i	$-0.686827\pi$
$547$	−25.9050	−1.10762	−0.553810	+	0.832643i	$0.686827\pi$
$548$	8.05446	0.344070
$549$	10.8181	0.461706
$550$	0	0
$551$	47.0552	2.00462
$552$	−6.76435	−0.287910
$553$	−3.60408	−0.153261
$554$	−4.77040	−0.202675
$555$	0	0
$556$	−12.8464	−0.544809
$557$	18.3250	0.776453	0.388227	−	0.921564i	$-0.373088\pi$
$557$	18.3250	0.776453	0.388227	+	0.921564i	$0.373088\pi$
$558$	4.79954	0.203181
$559$	0	0
$560$	0	0
$561$	−18.2657	−0.771178
$562$	−5.70927	−0.240831
$563$	−17.3172	−0.729832	−0.364916	−	0.931040i	$-0.618902\pi$
$563$	−17.3172	−0.729832	−0.364916	+	0.931040i	$0.618902\pi$
$564$	0.218381	0.00919551
$565$	0	0
$566$	21.2228	0.892061
$567$	−6.23900	−0.262013
$568$	1.36946	0.0574612
$569$	36.5689	1.53305	0.766524	−	0.642216i	$-0.221985\pi$
$569$	36.5689	1.53305	0.766524	+	0.642216i	$0.221985\pi$
$570$	0	0
$571$	31.7878	1.33028	0.665139	−	0.746720i	$-0.268372\pi$
$571$	31.7878	1.33028	0.665139	+	0.746720i	$0.268372\pi$
$572$	0	0
$573$	−15.5731	−0.650575
$574$	−0.407013	−0.0169884
$575$	0	0
$576$	−0.763924	−0.0318302
$577$	26.0002	1.08240	0.541202	−	0.840893i	$-0.317970\pi$
$577$	26.0002	1.08240	0.541202	+	0.840893i	$0.317970\pi$
$578$	8.47209	0.352393
$579$	−2.09743	−0.0871663
$580$	0	0
$581$	−4.73667	−0.196510
$582$	12.3722	0.512845
$583$	3.69333	0.152962
$584$	15.9009	0.657985
$585$	0	0
$586$	−23.9835	−0.990749
$587$	27.4016	1.13098	0.565492	−	0.824754i	$-0.308686\pi$
$587$	27.4016	1.13098	0.565492	+	0.824754i	$0.308686\pi$
$588$	8.91575	0.367679
$589$	−46.2728	−1.90664
$590$	0	0
$591$	−37.9005	−1.55902
$592$	1.80149	0.0740407
$593$	10.5439	0.432985	0.216492	−	0.976284i	$-0.430538\pi$
$593$	10.5439	0.432985	0.216492	+	0.976284i	$0.430538\pi$
$594$	13.6221	0.558922
$595$	0	0
$596$	11.6532	0.477335
$597$	−20.5806	−0.842307
$598$	0	0
$599$	−15.3122	−0.625640	−0.312820	−	0.949812i	$-0.601274\pi$
$599$	−15.3122	−0.625640	−0.312820	+	0.949812i	$0.601274\pi$
$600$	0	0
$601$	12.1851	0.497041	0.248521	−	0.968627i	$-0.420056\pi$
$601$	12.1851	0.497041	0.248521	+	0.968627i	$0.420056\pi$
$602$	−8.57531	−0.349504
$603$	5.03427	0.205011
$604$	−3.93672	−0.160183
$605$	0	0
$606$	29.8126	1.21105
$607$	18.3971	0.746716	0.373358	−	0.927687i	$-0.378206\pi$
$607$	18.3971	0.746716	0.373358	+	0.927687i	$0.378206\pi$
$608$	7.36506	0.298693
$609$	−9.73214	−0.394366
$610$	0	0
$611$	0	0
$612$	−3.85552	−0.155850
$613$	14.7497	0.595734	0.297867	−	0.954607i	$-0.403725\pi$
$613$	14.7497	0.595734	0.297867	+	0.954607i	$0.403725\pi$
$614$	17.0892	0.689666
$615$	0	0
$616$	2.46544	0.0993354
$617$	12.2578	0.493482	0.246741	−	0.969081i	$-0.420640\pi$
$617$	12.2578	0.493482	0.246741	+	0.969081i	$0.420640\pi$
$618$	−21.6380	−0.870407
$619$	−5.15917	−0.207365	−0.103682	−	0.994610i	$-0.533063\pi$
$619$	−5.15917	−0.207365	−0.103682	+	0.994610i	$0.533063\pi$
$620$	0	0
$621$	25.4605	1.02169
$622$	5.17968	0.207686
$623$	5.71354	0.228908
$624$	0	0
$625$	0	0
$626$	33.7809	1.35016
$627$	−26.6551	−1.06450
$628$	1.86131	0.0742742
$629$	9.09208	0.362525
$630$	0	0
$631$	11.4906	0.457432	0.228716	−	0.973493i	$-0.426547\pi$
$631$	11.4906	0.457432	0.228716	+	0.973493i	$0.426547\pi$
$632$	−3.53802	−0.140735
$633$	−14.6895	−0.583857
$634$	23.3968	0.929205
$635$	0	0
$636$	−2.28193	−0.0904842
$637$	0	0
$638$	15.4629	0.612183
$639$	−1.04616	−0.0413855
$640$	0	0
$641$	42.6494	1.68455	0.842275	−	0.539048i	$-0.181216\pi$
$641$	42.6494	1.68455	0.842275	+	0.539048i	$0.181216\pi$
$642$	−9.70571	−0.383054
$643$	−33.4244	−1.31813	−0.659064	−	0.752087i	$-0.729047\pi$
$643$	−33.4244	−1.31813	−0.659064	+	0.752087i	$0.729047\pi$
$644$	4.60805	0.181582
$645$	0	0
$646$	37.1714	1.46249
$647$	−21.1277	−0.830616	−0.415308	−	0.909681i	$-0.636326\pi$
$647$	−21.1277	−0.830616	−0.415308	+	0.909681i	$0.636326\pi$
$648$	−6.12465	−0.240599
$649$	7.83206	0.307435
$650$	0	0
$651$	9.57032	0.375090
$652$	−23.0167	−0.901402
$653$	23.4000	0.915711	0.457855	−	0.889027i	$-0.348618\pi$
$653$	23.4000	0.915711	0.457855	+	0.889027i	$0.348618\pi$
$654$	−20.7481	−0.811314
$655$	0	0
$656$	−0.399553	−0.0155999
$657$	−12.1471	−0.473903
$658$	−0.148767	−0.00579953
$659$	34.8524	1.35766	0.678829	−	0.734297i	$-0.262488\pi$
$659$	34.8524	1.35766	0.678829	+	0.734297i	$0.262488\pi$
$660$	0	0
$661$	−28.2985	−1.10068	−0.550342	−	0.834939i	$-0.685503\pi$
$661$	−28.2985	−1.10068	−0.550342	+	0.834939i	$0.685503\pi$
$662$	−5.46645	−0.212460
$663$	0	0
$664$	−4.64985	−0.180449
$665$	0	0
$666$	−1.37620	−0.0533266
$667$	28.9011	1.11905
$668$	6.11064	0.236428
$669$	−28.9441	−1.11904
$670$	0	0
$671$	−34.2737	−1.32312
$672$	−1.52327	−0.0587615
$673$	−12.7929	−0.493132	−0.246566	−	0.969126i	$-0.579302\pi$
$673$	−12.7929	−0.493132	−0.246566	+	0.969126i	$0.579302\pi$
$674$	25.4138	0.978902
$675$	0	0
$676$	0	0
$677$	31.8535	1.22423	0.612114	−	0.790769i	$-0.290319\pi$
$677$	31.8535	1.22423	0.612114	+	0.790769i	$0.290319\pi$
$678$	18.0287	0.692387
$679$	−8.42827	−0.323447
$680$	0	0
$681$	−31.0338	−1.18922
$682$	−15.2058	−0.582260
$683$	−51.5084	−1.97091	−0.985456	−	0.169928i	$-0.945646\pi$
$683$	−51.5084	−1.97091	−0.985456	+	0.169928i	$0.945646\pi$
$684$	−5.62635	−0.215129
$685$	0	0
$686$	−13.2043	−0.504144
$687$	34.8317	1.32891
$688$	−8.41814	−0.320938
$689$	0	0
$690$	0	0
$691$	−42.1525	−1.60356	−0.801778	−	0.597622i	$-0.796112\pi$
$691$	−42.1525	−1.60356	−0.801778	+	0.597622i	$0.796112\pi$
$692$	23.0755	0.877200
$693$	−1.88341	−0.0715448
$694$	−17.1079	−0.649408
$695$	0	0
$696$	−9.55376	−0.362134
$697$	−2.01654	−0.0763819
$698$	0.935233	0.0353991
$699$	−0.876230	−0.0331420
$700$	0	0
$701$	−31.6516	−1.19546	−0.597732	−	0.801696i	$-0.703931\pi$
$701$	−31.6516	−1.19546	−0.597732	+	0.801696i	$0.703931\pi$
$702$	0	0
$703$	13.2681	0.500414
$704$	2.42025	0.0912166
$705$	0	0
$706$	−0.243550	−0.00916612
$707$	−20.3091	−0.763802
$708$	−4.83904	−0.181862
$709$	24.5801	0.923126	0.461563	−	0.887108i	$-0.347289\pi$
$709$	24.5801	0.923126	0.461563	+	0.887108i	$0.347289\pi$
$710$	0	0
$711$	2.70278	0.101362
$712$	5.60882	0.210199
$713$	−28.4205	−1.06436
$714$	−7.68793	−0.287714
$715$	0	0
$716$	7.81690	0.292131
$717$	14.2958	0.533888
$718$	−14.7698	−0.551204
$719$	−4.11098	−0.153314	−0.0766568	−	0.997058i	$-0.524425\pi$
$719$	−4.11098	−0.153314	−0.0766568	+	0.997058i	$0.524425\pi$
$720$	0	0
$721$	14.7403	0.548959
$722$	35.2441	1.31165
$723$	17.2167	0.640298
$724$	−5.21467	−0.193802
$725$	0	0
$726$	7.68968	0.285391
$727$	−13.2521	−0.491492	−0.245746	−	0.969334i	$-0.579033\pi$
$727$	−13.2521	−0.491492	−0.245746	+	0.969334i	$0.579033\pi$
$728$	0	0
$729$	29.9280	1.10845
$730$	0	0
$731$	−42.4862	−1.57141
$732$	21.1760	0.782688
$733$	11.6585	0.430617	0.215309	−	0.976546i	$-0.430924\pi$
$733$	11.6585	0.430617	0.215309	+	0.976546i	$0.430924\pi$
$734$	−3.29260	−0.121532
$735$	0	0
$736$	4.52359	0.166742
$737$	−15.9495	−0.587506
$738$	0.305228	0.0112356
$739$	−33.7614	−1.24193	−0.620967	−	0.783836i	$-0.713260\pi$
$739$	−33.7614	−1.24193	−0.620967	+	0.783836i	$0.713260\pi$
$740$	0	0
$741$	0	0
$742$	1.55450	0.0570677
$743$	−32.1265	−1.17861	−0.589303	−	0.807912i	$-0.700598\pi$
$743$	−32.1265	−1.17861	−0.589303	+	0.807912i	$0.700598\pi$
$744$	9.39490	0.344434
$745$	0	0
$746$	−18.9067	−0.692222
$747$	3.55213	0.129966
$748$	12.2150	0.446624
$749$	6.61178	0.241589
$750$	0	0
$751$	40.3734	1.47325	0.736623	−	0.676304i	$-0.236419\pi$
$751$	40.3734	1.47325	0.736623	+	0.676304i	$0.236419\pi$
$752$	−0.146040	−0.00532553
$753$	22.4338	0.817534
$754$	0	0
$755$	0	0
$756$	5.73348	0.208525
$757$	−12.1790	−0.442652	−0.221326	−	0.975200i	$-0.571038\pi$
$757$	−12.1790	−0.442652	−0.221326	+	0.975200i	$0.571038\pi$
$758$	−13.7733	−0.500270
$759$	−16.3714	−0.594245
$760$	0	0
$761$	5.73188	0.207781	0.103890	−	0.994589i	$-0.466871\pi$
$761$	5.73188	0.207781	0.103890	+	0.994589i	$0.466871\pi$
$762$	−27.3592	−0.991119
$763$	14.1341	0.511689
$764$	10.4143	0.376777
$765$	0	0
$766$	−33.3243	−1.20405
$767$	0	0
$768$	−1.49535	−0.0539588
$769$	50.3927	1.81721	0.908603	−	0.417661i	$-0.137150\pi$
$769$	50.3927	1.81721	0.908603	+	0.417661i	$0.137150\pi$
$770$	0	0
$771$	1.34982	0.0486127
$772$	1.40263	0.0504819
$773$	31.9774	1.15015	0.575074	−	0.818102i	$-0.304973\pi$
$773$	31.9774	1.15015	0.575074	+	0.818102i	$0.304973\pi$
$774$	6.43082	0.231151
$775$	0	0
$776$	−8.27379	−0.297012
$777$	−2.74415	−0.0984459
$778$	−1.91117	−0.0685189
$779$	−2.94273	−0.105434
$780$	0	0
$781$	3.31443	0.118600
$782$	22.8305	0.816416
$783$	35.9596	1.28509
$784$	−5.96231	−0.212940
$785$	0	0
$786$	−32.6375	−1.16414
$787$	23.1697	0.825910	0.412955	−	0.910751i	$-0.364497\pi$
$787$	23.1697	0.825910	0.412955	+	0.910751i	$0.364497\pi$
$788$	25.3455	0.902897
$789$	26.2050	0.932921
$790$	0	0
$791$	−12.2816	−0.436683
$792$	−1.84889	−0.0656974
$793$	0	0
$794$	−13.5405	−0.480535
$795$	0	0
$796$	13.7630	0.487818
$797$	8.43667	0.298842	0.149421	−	0.988774i	$-0.452259\pi$
$797$	8.43667	0.298842	0.149421	+	0.988774i	$0.452259\pi$
$798$	−11.2190	−0.397148
$799$	−0.737063	−0.0260754
$800$	0	0
$801$	−4.28471	−0.151393
$802$	−28.9798	−1.02331
$803$	38.4842	1.35808
$804$	9.85438	0.347537
$805$	0	0
$806$	0	0
$807$	−4.75507	−0.167386
$808$	−19.9369	−0.701376
$809$	−11.8560	−0.416834	−0.208417	−	0.978040i	$-0.566831\pi$
$809$	−11.8560	−0.416834	−0.208417	+	0.978040i	$0.566831\pi$
$810$	0	0
$811$	−16.6388	−0.584268	−0.292134	−	0.956377i	$-0.594365\pi$
$811$	−16.6388	−0.584268	−0.292134	+	0.956377i	$0.594365\pi$
$812$	6.50826	0.228395
$813$	20.6381	0.723810
$814$	4.36005	0.152820
$815$	0	0
$816$	−7.54702	−0.264199
$817$	−62.0001	−2.16911
$818$	−9.77252	−0.341688
$819$	0	0
$820$	0	0
$821$	−25.6079	−0.893723	−0.446861	−	0.894603i	$-0.647458\pi$
$821$	−25.6079	−0.893723	−0.446861	+	0.894603i	$0.647458\pi$
$822$	−12.0442	−0.420091
$823$	24.1832	0.842974	0.421487	−	0.906834i	$-0.361508\pi$
$823$	24.1832	0.842974	0.421487	+	0.906834i	$0.361508\pi$
$824$	14.4702	0.504092
$825$	0	0
$826$	3.29647	0.114699
$827$	−15.5125	−0.539422	−0.269711	−	0.962941i	$-0.586928\pi$
$827$	−15.5125	−0.539422	−0.269711	+	0.962941i	$0.586928\pi$
$828$	−3.45568	−0.120093
$829$	47.4436	1.64779	0.823893	−	0.566746i	$-0.191798\pi$
$829$	47.4436	1.64779	0.823893	+	0.566746i	$0.191798\pi$
$830$	0	0
$831$	7.13342	0.247456
$832$	0	0
$833$	−30.0917	−1.04262
$834$	19.2099	0.665184
$835$	0	0
$836$	17.8253	0.616500
$837$	−35.3617	−1.22228
$838$	−24.9288	−0.861151
$839$	−38.9320	−1.34408	−0.672040	−	0.740514i	$-0.734582\pi$
$839$	−38.9320	−1.34408	−0.672040	+	0.740514i	$0.734582\pi$
$840$	0	0
$841$	11.8190	0.407552
$842$	12.7436	0.439175
$843$	8.53736	0.294042
$844$	9.82347	0.338138
$845$	0	0
$846$	0.111564	0.00383563
$847$	−5.23840	−0.179994
$848$	1.52601	0.0524035
$849$	−31.7356	−1.08916
$850$	0	0
$851$	8.14918	0.279350
$852$	−2.04782	−0.0701572
$853$	−11.8046	−0.404183	−0.202091	−	0.979367i	$-0.564774\pi$
$853$	−11.8046	−0.404183	−0.202091	+	0.979367i	$0.564774\pi$
$854$	−14.4256	−0.493635
$855$	0	0
$856$	6.49059	0.221844
$857$	26.7499	0.913759	0.456880	−	0.889529i	$-0.348967\pi$
$857$	26.7499	0.913759	0.456880	+	0.889529i	$0.348967\pi$
$858$	0	0
$859$	−21.2441	−0.724839	−0.362420	−	0.932015i	$-0.618049\pi$
$859$	−21.2441	−0.724839	−0.362420	+	0.932015i	$0.618049\pi$
$860$	0	0
$861$	0.608628	0.0207420
$862$	22.2564	0.758055
$863$	9.36613	0.318827	0.159413	−	0.987212i	$-0.449040\pi$
$863$	9.36613	0.318827	0.159413	+	0.987212i	$0.449040\pi$
$864$	5.62839	0.191482
$865$	0	0
$866$	30.4130	1.03348
$867$	−12.6688	−0.430254
$868$	−6.40004	−0.217232
$869$	−8.56290	−0.290476
$870$	0	0
$871$	0	0
$872$	13.8750	0.469868
$873$	6.32055	0.213918
$874$	33.3165	1.12695
$875$	0	0
$876$	−23.7775	−0.803366
$877$	−40.9589	−1.38308	−0.691542	−	0.722336i	$-0.743068\pi$
$877$	−40.9589	−1.38308	−0.691542	+	0.722336i	$0.743068\pi$
$878$	−12.5984	−0.425174
$879$	35.8637	1.20965
$880$	0	0
$881$	32.8926	1.10818	0.554090	−	0.832457i	$-0.313066\pi$
$881$	32.8926	1.10818	0.554090	+	0.832457i	$0.313066\pi$
$882$	4.55475	0.153366
$883$	28.2606	0.951044	0.475522	−	0.879704i	$-0.342259\pi$
$883$	28.2606	0.951044	0.475522	+	0.879704i	$0.342259\pi$
$884$	0	0
$885$	0	0
$886$	2.42243	0.0813833
$887$	6.76650	0.227197	0.113598	−	0.993527i	$-0.463762\pi$
$887$	6.76650	0.227197	0.113598	+	0.993527i	$0.463762\pi$
$888$	−2.69386	−0.0903999
$889$	18.6378	0.625091
$890$	0	0
$891$	−14.8232	−0.496595
$892$	19.3560	0.648088
$893$	−1.07559	−0.0359934
$894$	−17.4257	−0.582802
$895$	0	0
$896$	1.01867	0.0340314
$897$	0	0
$898$	−29.0963	−0.970954
$899$	−40.1403	−1.33875
$900$	0	0
$901$	7.70177	0.256583
$902$	−0.967019	−0.0321982
$903$	12.8231	0.426726
$904$	−12.0565	−0.400993
$905$	0	0
$906$	5.88677	0.195575
$907$	18.3605	0.609649	0.304824	−	0.952409i	$-0.401402\pi$
$907$	18.3605	0.609649	0.304824	+	0.952409i	$0.401402\pi$
$908$	20.7535	0.688729
$909$	15.2302	0.505155
$910$	0	0
$911$	−4.36340	−0.144566	−0.0722830	−	0.997384i	$-0.523028\pi$
$911$	−4.36340	−0.144566	−0.0722830	+	0.997384i	$0.523028\pi$
$912$	−11.0134	−0.364689
$913$	−11.2538	−0.372446
$914$	−5.25707	−0.173889
$915$	0	0
$916$	−23.2933	−0.769634
$917$	22.2335	0.734215
$918$	28.4064	0.937551
$919$	−42.7654	−1.41070	−0.705350	−	0.708859i	$-0.749210\pi$
$919$	−42.7654	−1.41070	−0.705350	+	0.708859i	$0.749210\pi$
$920$	0	0
$921$	−25.5544	−0.842047
$922$	21.6948	0.714479
$923$	0	0
$924$	−3.68670	−0.121283
$925$	0	0
$926$	−6.17734	−0.203000
$927$	−11.0541	−0.363064
$928$	6.38898	0.209728
$929$	22.2843	0.731124	0.365562	−	0.930787i	$-0.380877\pi$
$929$	22.2843	0.731124	0.365562	+	0.930787i	$0.380877\pi$
$930$	0	0
$931$	−43.9128	−1.43918
$932$	0.585969	0.0191941
$933$	−7.74545	−0.253575
$934$	9.01521	0.294987
$935$	0	0
$936$	0	0
$937$	28.0120	0.915113	0.457557	−	0.889180i	$-0.348725\pi$
$937$	28.0120	0.915113	0.457557	+	0.889180i	$0.348725\pi$
$938$	−6.71305	−0.219189
$939$	−50.5143	−1.64847
$940$	0	0
$941$	25.9180	0.844904	0.422452	−	0.906385i	$-0.361169\pi$
$941$	25.9180	0.844904	0.422452	+	0.906385i	$0.361169\pi$
$942$	−2.78331	−0.0906851
$943$	−1.80741	−0.0588575
$944$	3.23605	0.105325
$945$	0	0
$946$	−20.3740	−0.662416
$947$	−39.2544	−1.27560	−0.637798	−	0.770204i	$-0.720154\pi$
$947$	−39.2544	−1.27560	−0.637798	+	0.770204i	$0.720154\pi$
$948$	5.29059	0.171830
$949$	0	0
$950$	0	0
$951$	−34.9864	−1.13451
$952$	5.14122	0.166628
$953$	−38.9069	−1.26032	−0.630159	−	0.776466i	$-0.717010\pi$
$953$	−38.9069	−1.26032	−0.630159	+	0.776466i	$0.717010\pi$
$954$	−1.16576	−0.0377428
$955$	0	0
$956$	−9.56019	−0.309199
$957$	−23.1225	−0.747444
$958$	14.3406	0.463325
$959$	8.20485	0.264948
$960$	0	0
$961$	8.47282	0.273317
$962$	0	0
$963$	−4.95832	−0.159780
$964$	−11.5135	−0.370825
$965$	0	0
$966$	−6.89065	−0.221703
$967$	−49.1503	−1.58057	−0.790283	−	0.612742i	$-0.790066\pi$
$967$	−49.1503	−1.58057	−0.790283	+	0.612742i	$0.790066\pi$
$968$	−5.14239	−0.165283
$969$	−55.5843	−1.78562
$970$	0	0
$971$	2.29451	0.0736344	0.0368172	−	0.999322i	$-0.488278\pi$
$971$	2.29451	0.0736344	0.0368172	+	0.999322i	$0.488278\pi$
$972$	−7.72667	−0.247833
$973$	−13.0863	−0.419526
$974$	−15.5121	−0.497040
$975$	0	0
$976$	−14.1612	−0.453290
$977$	−40.4141	−1.29296	−0.646481	−	0.762930i	$-0.723760\pi$
$977$	−40.4141	−1.29296	−0.646481	+	0.762930i	$0.723760\pi$
$978$	34.4180	1.10057
$979$	13.5747	0.433851
$980$	0	0
$981$	−10.5995	−0.338415
$982$	2.60458	0.0831155
$983$	−5.97411	−0.190544	−0.0952722	−	0.995451i	$-0.530372\pi$
$983$	−5.97411	−0.190544	−0.0952722	+	0.995451i	$0.530372\pi$
$984$	0.597472	0.0190467
$985$	0	0
$986$	32.2451	1.02689
$987$	0.222459	0.00708094
$988$	0	0
$989$	−38.0802	−1.21088
$990$	0	0
$991$	−19.2849	−0.612604	−0.306302	−	0.951934i	$-0.599092\pi$
$991$	−19.2849	−0.612604	−0.306302	+	0.951934i	$0.599092\pi$
$992$	−6.28274	−0.199477
$993$	8.17426	0.259402
$994$	1.39503	0.0442476
$995$	0	0
$996$	6.95316	0.220319
$997$	52.5850	1.66538	0.832692	−	0.553736i	$-0.186798\pi$
$997$	52.5850	1.66538	0.832692	+	0.553736i	$0.186798\pi$
$998$	4.88458	0.154619
$999$	10.1395	0.320799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.da.1.2		9
5.2	odd	4		1690.2.b.g.339.17	yes	18
5.3	odd	4		1690.2.b.g.339.2	yes	18
5.4	even	2		8450.2.a.ct.1.8		9
13.12	even	2		8450.2.a.cw.1.2		9
65.8	even	4		1690.2.c.h.1689.5		18
65.12	odd	4		1690.2.b.f.339.8	✓	18
65.18	even	4		1690.2.c.g.1689.5		18
65.38	odd	4		1690.2.b.f.339.11	yes	18
65.47	even	4		1690.2.c.g.1689.14		18
65.57	even	4		1690.2.c.h.1689.14		18
65.64	even	2		8450.2.a.cx.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.8	✓	18	65.12	odd	4
1690.2.b.f.339.11	yes	18	65.38	odd	4
1690.2.b.g.339.2	yes	18	5.3	odd	4
1690.2.b.g.339.17	yes	18	5.2	odd	4
1690.2.c.g.1689.5		18	65.18	even	4
1690.2.c.g.1689.14		18	65.47	even	4
1690.2.c.h.1689.5		18	65.8	even	4
1690.2.c.h.1689.14		18	65.57	even	4
8450.2.a.ct.1.8		9	5.4	even	2
8450.2.a.cw.1.2		9	13.12	even	2
8450.2.a.cx.1.8		9	65.64	even	2
8450.2.a.da.1.2		9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.49535 q^{3} +1.00000 q^{4} -1.49535 q^{6} +1.01867 q^{7} +1.00000 q^{8} -0.763924 q^{9} +2.42025 q^{11} -1.49535 q^{12} +1.01867 q^{14} +1.00000 q^{16} +5.04699 q^{17} -0.763924 q^{18} +7.36506 q^{19} -1.52327 q^{21} +2.42025 q^{22} +4.52359 q^{23} -1.49535 q^{24} +5.62839 q^{27} +1.01867 q^{28} +6.38898 q^{29} -6.28274 q^{31} +1.00000 q^{32} -3.61912 q^{33} +5.04699 q^{34} -0.763924 q^{36} +1.80149 q^{37} +7.36506 q^{38} -0.399553 q^{41} -1.52327 q^{42} -8.41814 q^{43} +2.42025 q^{44} +4.52359 q^{46} -0.146040 q^{47} -1.49535 q^{48} -5.96231 q^{49} -7.54702 q^{51} +1.52601 q^{53} +5.62839 q^{54} +1.01867 q^{56} -11.0134 q^{57} +6.38898 q^{58} +3.23605 q^{59} -14.1612 q^{61} -6.28274 q^{62} -0.778187 q^{63} +1.00000 q^{64} -3.61912 q^{66} -6.59001 q^{67} +5.04699 q^{68} -6.76435 q^{69} +1.36946 q^{71} -0.763924 q^{72} +15.9009 q^{73} +1.80149 q^{74} +7.36506 q^{76} +2.46544 q^{77} -3.53802 q^{79} -6.12465 q^{81} -0.399553 q^{82} -4.64985 q^{83} -1.52327 q^{84} -8.41814 q^{86} -9.55376 q^{87} +2.42025 q^{88} +5.60882 q^{89} +4.52359 q^{92} +9.39490 q^{93} -0.146040 q^{94} -1.49535 q^{96} -8.27379 q^{97} -5.96231 q^{98} -1.84889 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + 4 q^{11} + 7 q^{12} + q^{14} + 9 q^{16} + 12 q^{17} + 8 q^{18} + 6 q^{19} + 8 q^{21} + 4 q^{22} + 11 q^{23} + 7 q^{24} + 34 q^{27}+ \cdots + 16 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + 7 * q^3 + 9 * q^4 + 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 + 4 * q^11 + 7 * q^12 + q^14 + 9 * q^16 + 12 * q^17 + 8 * q^18 + 6 * q^19 + 8 * q^21 + 4 * q^22 + 11 * q^23 + 7 * q^24 + 34 * q^27 + q^28 + 15 * q^29 - 6 * q^31 + 9 * q^32 - 14 * q^33 + 12 * q^34 + 8 * q^36 - 10 * q^37 + 6 * q^38 + 27 * q^41 + 8 * q^42 + 41 * q^43 + 4 * q^44 + 11 * q^46 + 5 * q^47 + 7 * q^48 - 10 * q^49 + 12 * q^51 + 18 * q^53 + 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 + 20 * q^59 - 19 * q^61 - 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 + 12 * q^68 - 14 * q^69 - 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 + 6 * q^76 + 30 * q^77 - 14 * q^79 + 53 * q^81 + 27 * q^82 + q^83 + 8 * q^84 + 41 * q^86 + 33 * q^87 + 4 * q^88 + 43 * q^89 + 11 * q^92 - 28 * q^93 + 5 * q^94 + 7 * q^96 + 6 * q^97 - 10 * q^98 + 16 * q^99

Introduction

L-functions

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