LMFDB - Embedded newform 8450.2.a.bn.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,-3,3,0,3,4,-3,8,0,3,-3,0,-4,0,3,-5] 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 338)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.04892 q^{3} +1.00000 q^{4} -2.04892 q^{6} +1.10992 q^{7} -1.00000 q^{8} +1.19806 q^{9} +2.35690 q^{11} +2.04892 q^{12} -1.10992 q^{14} +1.00000 q^{16} -5.96077 q^{17} -1.19806 q^{18} -0.911854 q^{19} +2.27413 q^{21} -2.35690 q^{22} -3.38404 q^{23} -2.04892 q^{24} -3.69202 q^{27} +1.10992 q^{28} -3.78017 q^{29} -8.49396 q^{31} -1.00000 q^{32} +4.82908 q^{33} +5.96077 q^{34} +1.19806 q^{36} +4.89008 q^{37} +0.911854 q^{38} +7.18598 q^{41} -2.27413 q^{42} +0.515729 q^{43} +2.35690 q^{44} +3.38404 q^{46} +6.98792 q^{47} +2.04892 q^{48} -5.76809 q^{49} -12.2131 q^{51} +3.38404 q^{53} +3.69202 q^{54} -1.10992 q^{56} -1.86831 q^{57} +3.78017 q^{58} -10.1468 q^{59} -0.439665 q^{61} +8.49396 q^{62} +1.32975 q^{63} +1.00000 q^{64} -4.82908 q^{66} +2.14675 q^{67} -5.96077 q^{68} -6.93362 q^{69} -0.615957 q^{71} -1.19806 q^{72} +6.32304 q^{73} -4.89008 q^{74} -0.911854 q^{76} +2.61596 q^{77} -15.4819 q^{79} -11.1588 q^{81} -7.18598 q^{82} -0.911854 q^{83} +2.27413 q^{84} -0.515729 q^{86} -7.74525 q^{87} -2.35690 q^{88} -3.75063 q^{89} -3.38404 q^{92} -17.4034 q^{93} -6.98792 q^{94} -2.04892 q^{96} -14.6746 q^{97} +5.76809 q^{98} +2.82371 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{11} - 3 q^{12} - 4 q^{14} + 3 q^{16} - 5 q^{17} - 8 q^{18} + q^{19} - 4 q^{21} - 3 q^{22} + 3 q^{24} - 6 q^{27} + 4 q^{28}+ \cdots + q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^11 - 3 * q^12 - 4 * q^14 + 3 * q^16 - 5 * q^17 - 8 * q^18 + q^19 - 4 * q^21 - 3 * q^22 + 3 * q^24 - 6 * q^27 + 4 * q^28 - 10 * q^29 - 16 * q^31 - 3 * q^32 + 4 * q^33 + 5 * q^34 + 8 * q^36 + 14 * q^37 - q^38 + 7 * q^41 + 4 * q^42 - 11 * q^43 + 3 * q^44 + 2 * q^47 - 3 * q^48 + 3 * q^49 - 16 * q^51 + 6 * q^54 - 4 * q^56 - 8 * q^57 + 10 * q^58 - 3 * q^59 - 4 * q^61 + 16 * q^62 + 6 * q^63 + 3 * q^64 - 4 * q^66 - 21 * q^67 - 5 * q^68 - 14 * q^69 - 12 * q^71 - 8 * q^72 - q^73 - 14 * q^74 + q^76 + 18 * q^77 - 18 * q^79 - 25 * q^81 - 7 * q^82 + q^83 - 4 * q^84 + 11 * q^86 + 10 * q^87 - 3 * q^88 + 25 * q^89 + 2 * q^93 - 2 * q^94 + 3 * q^96 - 23 * q^97 - 3 * q^98 + 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
$3$	2.04892	1.18294	0.591471	−	0.806326i	$-0.298547\pi$
$3$	2.04892	1.18294	0.591471	+	0.806326i	$0.298547\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.04892	−0.836467
$7$	1.10992	0.419509	0.209754	−	0.977754i	$-0.432734\pi$
$7$	1.10992	0.419509	0.209754	+	0.977754i	$0.432734\pi$
$8$	−1.00000	−0.353553
$9$	1.19806	0.399354
$10$	0	0
$11$	2.35690	0.710631	0.355315	−	0.934746i	$-0.384373\pi$
$11$	2.35690	0.710631	0.355315	+	0.934746i	$0.384373\pi$
$12$	2.04892	0.591471
$13$	0	0
$13$	0	0
$14$	−1.10992	−0.296638
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.96077	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$17$	−5.96077	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$18$	−1.19806	−0.282386
$19$	−0.911854	−0.209194	−0.104597	−	0.994515i	$-0.533355\pi$
$19$	−0.911854	−0.209194	−0.104597	+	0.994515i	$0.533355\pi$
$20$	0	0
$21$	2.27413	0.496255
$22$	−2.35690	−0.502492
$23$	−3.38404	−0.705622	−0.352811	−	0.935695i	$-0.614774\pi$
$23$	−3.38404	−0.705622	−0.352811	+	0.935695i	$0.614774\pi$
$24$	−2.04892	−0.418234
$25$	0	0
$26$	0	0
$27$	−3.69202	−0.710530
$28$	1.10992	0.209754
$29$	−3.78017	−0.701959	−0.350980	−	0.936383i	$-0.614151\pi$
$29$	−3.78017	−0.701959	−0.350980	+	0.936383i	$0.614151\pi$
$30$	0	0
$31$	−8.49396	−1.52556	−0.762780	−	0.646658i	$-0.776166\pi$
$31$	−8.49396	−1.52556	−0.762780	+	0.646658i	$0.776166\pi$
$32$	−1.00000	−0.176777
$33$	4.82908	0.840636
$34$	5.96077	1.02226
$35$	0	0
$36$	1.19806	0.199677
$37$	4.89008	0.803925	0.401962	−	0.915656i	$-0.368328\pi$
$37$	4.89008	0.803925	0.401962	+	0.915656i	$0.368328\pi$
$38$	0.911854	0.147922
$39$	0	0
$40$	0	0
$41$	7.18598	1.12226	0.561131	−	0.827727i	$-0.310366\pi$
$41$	7.18598	1.12226	0.561131	+	0.827727i	$0.310366\pi$
$42$	−2.27413	−0.350905
$43$	0.515729	0.0786480	0.0393240	−	0.999227i	$-0.487480\pi$
$43$	0.515729	0.0786480	0.0393240	+	0.999227i	$0.487480\pi$
$44$	2.35690	0.355315
$45$	0	0
$46$	3.38404	0.498950
$47$	6.98792	1.01929	0.509646	−	0.860384i	$-0.329776\pi$
$47$	6.98792	1.01929	0.509646	+	0.860384i	$0.329776\pi$
$48$	2.04892	0.295736
$49$	−5.76809	−0.824012
$50$	0	0
$51$	−12.2131	−1.71018
$52$	0	0
$53$	3.38404	0.464834	0.232417	−	0.972616i	$-0.425337\pi$
$53$	3.38404	0.464834	0.232417	+	0.972616i	$0.425337\pi$
$54$	3.69202	0.502420
$55$	0	0
$56$	−1.10992	−0.148319
$57$	−1.86831	−0.247464
$58$	3.78017	0.496360
$59$	−10.1468	−1.32099	−0.660497	−	0.750828i	$-0.729655\pi$
$59$	−10.1468	−1.32099	−0.660497	+	0.750828i	$0.729655\pi$
$60$	0	0
$61$	−0.439665	−0.0562933	−0.0281467	−	0.999604i	$-0.508961\pi$
$61$	−0.439665	−0.0562933	−0.0281467	+	0.999604i	$0.508961\pi$
$62$	8.49396	1.07873
$63$	1.32975	0.167533
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.82908	−0.594419
$67$	2.14675	0.262268	0.131134	−	0.991365i	$-0.458138\pi$
$67$	2.14675	0.262268	0.131134	+	0.991365i	$0.458138\pi$
$68$	−5.96077	−0.722850
$69$	−6.93362	−0.834710
$70$	0	0
$71$	−0.615957	−0.0731007	−0.0365503	−	0.999332i	$-0.511637\pi$
$71$	−0.615957	−0.0731007	−0.0365503	+	0.999332i	$0.511637\pi$
$72$	−1.19806	−0.141193
$73$	6.32304	0.740056	0.370028	−	0.929021i	$-0.379348\pi$
$73$	6.32304	0.740056	0.370028	+	0.929021i	$0.379348\pi$
$74$	−4.89008	−0.568461
$75$	0	0
$76$	−0.911854	−0.104597
$77$	2.61596	0.298116
$78$	0	0
$79$	−15.4819	−1.74185	−0.870924	−	0.491418i	$-0.836479\pi$
$79$	−15.4819	−1.74185	−0.870924	+	0.491418i	$0.836479\pi$
$80$	0	0
$81$	−11.1588	−1.23987
$82$	−7.18598	−0.793559
$83$	−0.911854	−0.100089	−0.0500445	−	0.998747i	$-0.515936\pi$
$83$	−0.911854	−0.100089	−0.0500445	+	0.998747i	$0.515936\pi$
$84$	2.27413	0.248128
$85$	0	0
$86$	−0.515729	−0.0556125
$87$	−7.74525	−0.830378
$88$	−2.35690	−0.251246
$89$	−3.75063	−0.397566	−0.198783	−	0.980044i	$-0.563699\pi$
$89$	−3.75063	−0.397566	−0.198783	+	0.980044i	$0.563699\pi$
$90$	0	0
$91$	0	0
$92$	−3.38404	−0.352811
$93$	−17.4034	−1.80465
$94$	−6.98792	−0.720749
$95$	0	0
$96$	−2.04892	−0.209117
$97$	−14.6746	−1.48998	−0.744988	−	0.667078i	$-0.767545\pi$
$97$	−14.6746	−1.48998	−0.744988	+	0.667078i	$0.767545\pi$
$98$	5.76809	0.582665
$99$	2.82371	0.283793
$100$	0	0
$101$	8.76809	0.872457	0.436229	−	0.899836i	$-0.356314\pi$
$101$	8.76809	0.872457	0.436229	+	0.899836i	$0.356314\pi$
$102$	12.2131	1.20928
$103$	−18.8116	−1.85356	−0.926782	−	0.375599i	$-0.877437\pi$
$103$	−18.8116	−1.85356	−0.926782	+	0.375599i	$0.877437\pi$
$104$	0	0
$105$	0	0
$106$	−3.38404	−0.328687
$107$	−18.0519	−1.74514	−0.872572	−	0.488486i	$-0.837549\pi$
$107$	−18.0519	−1.74514	−0.872572	+	0.488486i	$0.837549\pi$
$108$	−3.69202	−0.355265
$109$	−6.09783	−0.584067	−0.292033	−	0.956408i	$-0.594332\pi$
$109$	−6.09783	−0.584067	−0.292033	+	0.956408i	$0.594332\pi$
$110$	0	0
$111$	10.0194	0.950997
$112$	1.10992	0.104877
$113$	12.2010	1.14778	0.573889	−	0.818933i	$-0.305434\pi$
$113$	12.2010	1.14778	0.573889	+	0.818933i	$0.305434\pi$
$114$	1.86831	0.174984
$115$	0	0
$116$	−3.78017	−0.350980
$117$	0	0
$118$	10.1468	0.934084
$119$	−6.61596	−0.606484
$120$	0	0
$121$	−5.44504	−0.495004
$122$	0.439665	0.0398054
$123$	14.7235	1.32757
$124$	−8.49396	−0.762780
$125$	0	0
$126$	−1.32975	−0.118463
$127$	−11.4276	−1.01403	−0.507017	−	0.861936i	$-0.669252\pi$
$127$	−11.4276	−1.01403	−0.507017	+	0.861936i	$0.669252\pi$
$128$	−1.00000	−0.0883883
$129$	1.05669	0.0930361
$130$	0	0
$131$	2.29590	0.200593	0.100297	−	0.994958i	$-0.468021\pi$
$131$	2.29590	0.200593	0.100297	+	0.994958i	$0.468021\pi$
$132$	4.82908	0.420318
$133$	−1.01208	−0.0877586
$134$	−2.14675	−0.185451
$135$	0	0
$136$	5.96077	0.511132
$137$	9.08038	0.775789	0.387894	−	0.921704i	$-0.373203\pi$
$137$	9.08038	0.775789	0.387894	+	0.921704i	$0.373203\pi$
$138$	6.93362	0.590229
$139$	18.9051	1.60351	0.801757	−	0.597650i	$-0.203899\pi$
$139$	18.9051	1.60351	0.801757	+	0.597650i	$0.203899\pi$
$140$	0	0
$141$	14.3177	1.20577
$142$	0.615957	0.0516900
$143$	0	0
$144$	1.19806	0.0998385
$145$	0	0
$146$	−6.32304	−0.523299
$147$	−11.8183	−0.974760
$148$	4.89008	0.401962
$149$	−18.6896	−1.53111	−0.765557	−	0.643368i	$-0.777537\pi$
$149$	−18.6896	−1.53111	−0.765557	+	0.643368i	$0.777537\pi$
$150$	0	0
$151$	0.317667	0.0258514	0.0129257	−	0.999916i	$-0.495886\pi$
$151$	0.317667	0.0258514	0.0129257	+	0.999916i	$0.495886\pi$
$152$	0.911854	0.0739611
$153$	−7.14138	−0.577346
$154$	−2.61596	−0.210800
$155$	0	0
$156$	0	0
$157$	−18.8901	−1.50759	−0.753796	−	0.657108i	$-0.771780\pi$
$157$	−18.8901	−1.50759	−0.753796	+	0.657108i	$0.771780\pi$
$158$	15.4819	1.23167
$159$	6.93362	0.549872
$160$	0	0
$161$	−3.75600	−0.296015
$162$	11.1588	0.876721
$163$	−4.33273	−0.339366	−0.169683	−	0.985499i	$-0.554274\pi$
$163$	−4.33273	−0.339366	−0.169683	+	0.985499i	$0.554274\pi$
$164$	7.18598	0.561131
$165$	0	0
$166$	0.911854	0.0707736
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	−2.27413	−0.175453
$169$	0	0
$170$	0	0
$171$	−1.09246	−0.0835423
$172$	0.515729	0.0393240
$173$	10.9879	0.835396	0.417698	−	0.908586i	$-0.362837\pi$
$173$	10.9879	0.835396	0.417698	+	0.908586i	$0.362837\pi$
$174$	7.74525	0.587166
$175$	0	0
$176$	2.35690	0.177658
$177$	−20.7899	−1.56266
$178$	3.75063	0.281121
$179$	4.65519	0.347945	0.173972	−	0.984751i	$-0.444340\pi$
$179$	4.65519	0.347945	0.173972	+	0.984751i	$0.444340\pi$
$180$	0	0
$181$	−1.06638	−0.0792631	−0.0396315	−	0.999214i	$-0.512618\pi$
$181$	−1.06638	−0.0792631	−0.0396315	+	0.999214i	$0.512618\pi$
$182$	0	0
$183$	−0.900837	−0.0665918
$184$	3.38404	0.249475
$185$	0	0
$186$	17.4034	1.27608
$187$	−14.0489	−1.02736
$188$	6.98792	0.509646
$189$	−4.09783	−0.298074
$190$	0	0
$191$	0.890084	0.0644042	0.0322021	−	0.999481i	$-0.489748\pi$
$191$	0.890084	0.0644042	0.0322021	+	0.999481i	$0.489748\pi$
$192$	2.04892	0.147868
$193$	16.2174	1.16736	0.583678	−	0.811985i	$-0.301613\pi$
$193$	16.2174	1.16736	0.583678	+	0.811985i	$0.301613\pi$
$194$	14.6746	1.05357
$195$	0	0
$196$	−5.76809	−0.412006
$197$	−11.4711	−0.817284	−0.408642	−	0.912695i	$-0.633997\pi$
$197$	−11.4711	−0.817284	−0.408642	+	0.912695i	$0.633997\pi$
$198$	−2.82371	−0.200672
$199$	−3.79954	−0.269343	−0.134671	−	0.990890i	$-0.542998\pi$
$199$	−3.79954	−0.269343	−0.134671	+	0.990890i	$0.542998\pi$
$200$	0	0
$201$	4.39852	0.310247
$202$	−8.76809	−0.616920
$203$	−4.19567	−0.294478
$204$	−12.2131	−0.855090
$205$	0	0
$206$	18.8116	1.31067
$207$	−4.05429	−0.281793
$208$	0	0
$209$	−2.14914	−0.148659
$210$	0	0
$211$	−25.0465	−1.72427	−0.862137	−	0.506675i	$-0.830874\pi$
$211$	−25.0465	−1.72427	−0.862137	+	0.506675i	$0.830874\pi$
$212$	3.38404	0.232417
$213$	−1.26205	−0.0864739
$214$	18.0519	1.23400
$215$	0	0
$216$	3.69202	0.251210
$217$	−9.42758	−0.639986
$218$	6.09783	0.412997
$219$	12.9554	0.875444
$220$	0	0
$221$	0	0
$222$	−10.0194	−0.672457
$223$	12.9879	0.869735	0.434868	−	0.900494i	$-0.356795\pi$
$223$	12.9879	0.869735	0.434868	+	0.900494i	$0.356795\pi$
$224$	−1.10992	−0.0741594
$225$	0	0
$226$	−12.2010	−0.811602
$227$	−13.8049	−0.916265	−0.458132	−	0.888884i	$-0.651481\pi$
$227$	−13.8049	−0.916265	−0.458132	+	0.888884i	$0.651481\pi$
$228$	−1.86831	−0.123732
$229$	11.5603	0.763928	0.381964	−	0.924177i	$-0.375248\pi$
$229$	11.5603	0.763928	0.381964	+	0.924177i	$0.375248\pi$
$230$	0	0
$231$	5.35988	0.352654
$232$	3.78017	0.248180
$233$	−9.77479	−0.640368	−0.320184	−	0.947355i	$-0.603745\pi$
$233$	−9.77479	−0.640368	−0.320184	+	0.947355i	$0.603745\pi$
$234$	0	0
$235$	0	0
$236$	−10.1468	−0.660497
$237$	−31.7211	−2.06051
$238$	6.61596	0.428849
$239$	0.944378	0.0610867	0.0305434	−	0.999533i	$-0.490276\pi$
$239$	0.944378	0.0610867	0.0305434	+	0.999533i	$0.490276\pi$
$240$	0	0
$241$	0.219833	0.0141607	0.00708033	−	0.999975i	$-0.497746\pi$
$241$	0.219833	0.0141607	0.00708033	+	0.999975i	$0.497746\pi$
$242$	5.44504	0.350021
$243$	−11.7875	−0.756166
$244$	−0.439665	−0.0281467
$245$	0	0
$246$	−14.7235	−0.938735
$247$	0	0
$248$	8.49396	0.539367
$249$	−1.86831	−0.118400
$250$	0	0
$251$	16.2543	1.02596	0.512980	−	0.858400i	$-0.328541\pi$
$251$	16.2543	1.02596	0.512980	+	0.858400i	$0.328541\pi$
$252$	1.32975	0.0837663
$253$	−7.97584	−0.501437
$254$	11.4276	0.717030
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.4373	1.39960	0.699799	−	0.714340i	$-0.253273\pi$
$257$	22.4373	1.39960	0.699799	+	0.714340i	$0.253273\pi$
$258$	−1.05669	−0.0657865
$259$	5.42758	0.337254
$260$	0	0
$261$	−4.52888	−0.280330
$262$	−2.29590	−0.141841
$263$	10.4940	0.647085	0.323543	−	0.946214i	$-0.395126\pi$
$263$	10.4940	0.647085	0.323543	+	0.946214i	$0.395126\pi$
$264$	−4.82908	−0.297210
$265$	0	0
$266$	1.01208	0.0620547
$267$	−7.68473	−0.470298
$268$	2.14675	0.131134
$269$	26.4155	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$269$	26.4155	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$270$	0	0
$271$	22.0301	1.33824	0.669118	−	0.743157i	$-0.266672\pi$
$271$	22.0301	1.33824	0.669118	+	0.743157i	$0.266672\pi$
$272$	−5.96077	−0.361425
$273$	0	0
$274$	−9.08038	−0.548566
$275$	0	0
$276$	−6.93362	−0.417355
$277$	−2.17629	−0.130761	−0.0653804	−	0.997860i	$-0.520826\pi$
$277$	−2.17629	−0.130761	−0.0653804	+	0.997860i	$0.520826\pi$
$278$	−18.9051	−1.13386
$279$	−10.1763	−0.609239
$280$	0	0
$281$	−25.0030	−1.49155	−0.745776	−	0.666196i	$-0.767921\pi$
$281$	−25.0030	−1.49155	−0.745776	+	0.666196i	$0.767921\pi$
$282$	−14.3177	−0.852605
$283$	16.3153	0.969842	0.484921	−	0.874558i	$-0.338848\pi$
$283$	16.3153	0.969842	0.484921	+	0.874558i	$0.338848\pi$
$284$	−0.615957	−0.0365503
$285$	0	0
$286$	0	0
$287$	7.97584	0.470799
$288$	−1.19806	−0.0705965
$289$	18.5308	1.09005
$290$	0	0
$291$	−30.0670	−1.76256
$292$	6.32304	0.370028
$293$	−1.87800	−0.109714	−0.0548570	−	0.998494i	$-0.517470\pi$
$293$	−1.87800	−0.109714	−0.0548570	+	0.998494i	$0.517470\pi$
$294$	11.8183	0.689259
$295$	0	0
$296$	−4.89008	−0.284230
$297$	−8.70171	−0.504924
$298$	18.6896	1.08266
$299$	0	0
$300$	0	0
$301$	0.572417	0.0329935
$302$	−0.317667	−0.0182797
$303$	17.9651	1.03207
$304$	−0.911854	−0.0522984
$305$	0	0
$306$	7.14138	0.408245
$307$	−23.9801	−1.36862	−0.684310	−	0.729192i	$-0.739896\pi$
$307$	−23.9801	−1.36862	−0.684310	+	0.729192i	$0.739896\pi$
$308$	2.61596	0.149058
$309$	−38.5435	−2.19266
$310$	0	0
$311$	−5.38404	−0.305301	−0.152651	−	0.988280i	$-0.548781\pi$
$311$	−5.38404	−0.305301	−0.152651	+	0.988280i	$0.548781\pi$
$312$	0	0
$313$	−18.9487	−1.07104	−0.535522	−	0.844522i	$-0.679885\pi$
$313$	−18.9487	−1.07104	−0.535522	+	0.844522i	$0.679885\pi$
$314$	18.8901	1.06603
$315$	0	0
$316$	−15.4819	−0.870924
$317$	−11.5013	−0.645975	−0.322987	−	0.946403i	$-0.604687\pi$
$317$	−11.5013	−0.645975	−0.322987	+	0.946403i	$0.604687\pi$
$318$	−6.93362	−0.388818
$319$	−8.90946	−0.498834
$320$	0	0
$321$	−36.9869	−2.06440
$322$	3.75600	0.209314
$323$	5.43535	0.302431
$324$	−11.1588	−0.619935
$325$	0	0
$326$	4.33273	0.239968
$327$	−12.4940	−0.690918
$328$	−7.18598	−0.396779
$329$	7.75600	0.427602
$330$	0	0
$331$	34.6112	1.90240	0.951201	−	0.308572i	$-0.0998511\pi$
$331$	34.6112	1.90240	0.951201	+	0.308572i	$0.0998511\pi$
$332$	−0.911854	−0.0500445
$333$	5.85862	0.321051
$334$	−14.0000	−0.766046
$335$	0	0
$336$	2.27413	0.124064
$337$	−1.95407	−0.106445	−0.0532224	−	0.998583i	$-0.516949\pi$
$337$	−1.95407	−0.106445	−0.0532224	+	0.998583i	$0.516949\pi$
$338$	0	0
$339$	24.9989	1.35776
$340$	0	0
$341$	−20.0194	−1.08411
$342$	1.09246	0.0590734
$343$	−14.1715	−0.765189
$344$	−0.515729	−0.0278063
$345$	0	0
$346$	−10.9879	−0.590714
$347$	6.41550	0.344402	0.172201	−	0.985062i	$-0.444912\pi$
$347$	6.41550	0.344402	0.172201	+	0.985062i	$0.444912\pi$
$348$	−7.74525	−0.415189
$349$	1.08575	0.0581190	0.0290595	−	0.999578i	$-0.490749\pi$
$349$	1.08575	0.0581190	0.0290595	+	0.999578i	$0.490749\pi$
$350$	0	0
$351$	0	0
$352$	−2.35690	−0.125623
$353$	−4.28919	−0.228291	−0.114145	−	0.993464i	$-0.536413\pi$
$353$	−4.28919	−0.228291	−0.114145	+	0.993464i	$0.536413\pi$
$354$	20.7899	1.10497
$355$	0	0
$356$	−3.75063	−0.198783
$357$	−13.5555	−0.717436
$358$	−4.65519	−0.246034
$359$	−15.5060	−0.818378	−0.409189	−	0.912450i	$-0.634188\pi$
$359$	−15.5060	−0.818378	−0.409189	+	0.912450i	$0.634188\pi$
$360$	0	0
$361$	−18.1685	−0.956238
$362$	1.06638	0.0560475
$363$	−11.1564	−0.585561
$364$	0	0
$365$	0	0
$366$	0.900837	0.0470875
$367$	−17.4276	−0.909712	−0.454856	−	0.890565i	$-0.650309\pi$
$367$	−17.4276	−0.909712	−0.454856	+	0.890565i	$0.650309\pi$
$368$	−3.38404	−0.176405
$369$	8.60925	0.448180
$370$	0	0
$371$	3.75600	0.195002
$372$	−17.4034	−0.902325
$373$	−8.19567	−0.424356	−0.212178	−	0.977231i	$-0.568056\pi$
$373$	−8.19567	−0.424356	−0.212178	+	0.977231i	$0.568056\pi$
$374$	14.0489	0.726452
$375$	0	0
$376$	−6.98792	−0.360374
$377$	0	0
$378$	4.09783	0.210770
$379$	−15.0476	−0.772943	−0.386471	−	0.922301i	$-0.626306\pi$
$379$	−15.0476	−0.772943	−0.386471	+	0.922301i	$0.626306\pi$
$380$	0	0
$381$	−23.4142	−1.19954
$382$	−0.890084	−0.0455406
$383$	11.1207	0.568240	0.284120	−	0.958789i	$-0.408299\pi$
$383$	11.1207	0.568240	0.284120	+	0.958789i	$0.408299\pi$
$384$	−2.04892	−0.104558
$385$	0	0
$386$	−16.2174	−0.825446
$387$	0.617876	0.0314084
$388$	−14.6746	−0.744988
$389$	−8.04354	−0.407824	−0.203912	−	0.978989i	$-0.565366\pi$
$389$	−8.04354	−0.407824	−0.203912	+	0.978989i	$0.565366\pi$
$390$	0	0
$391$	20.1715	1.02012
$392$	5.76809	0.291332
$393$	4.70410	0.237291
$394$	11.4711	0.577907
$395$	0	0
$396$	2.82371	0.141897
$397$	21.9081	1.09954	0.549769	−	0.835317i	$-0.314716\pi$
$397$	21.9081	1.09954	0.549769	+	0.835317i	$0.314716\pi$
$398$	3.79954	0.190454
$399$	−2.07367	−0.103813
$400$	0	0
$401$	17.4426	0.871044	0.435522	−	0.900178i	$-0.356564\pi$
$401$	17.4426	0.871044	0.435522	+	0.900178i	$0.356564\pi$
$402$	−4.39852	−0.219378
$403$	0	0
$404$	8.76809	0.436229
$405$	0	0
$406$	4.19567	0.208228
$407$	11.5254	0.571294
$408$	12.2131	0.604640
$409$	−17.4330	−0.862004	−0.431002	−	0.902351i	$-0.641840\pi$
$409$	−17.4330	−0.862004	−0.431002	+	0.902351i	$0.641840\pi$
$410$	0	0
$411$	18.6049	0.917714
$412$	−18.8116	−0.926782
$413$	−11.2620	−0.554169
$414$	4.05429	0.199258
$415$	0	0
$416$	0	0
$417$	38.7351	1.89687
$418$	2.14914	0.105118
$419$	9.97584	0.487352	0.243676	−	0.969857i	$-0.421647\pi$
$419$	9.97584	0.487352	0.243676	+	0.969857i	$0.421647\pi$
$420$	0	0
$421$	0.615957	0.0300199	0.0150100	−	0.999887i	$-0.495222\pi$
$421$	0.615957	0.0300199	0.0150100	+	0.999887i	$0.495222\pi$
$422$	25.0465	1.21925
$423$	8.37196	0.407059
$424$	−3.38404	−0.164344
$425$	0	0
$426$	1.26205	0.0611463
$427$	−0.487991	−0.0236156
$428$	−18.0519	−0.872572
$429$	0	0
$430$	0	0
$431$	−14.7922	−0.712518	−0.356259	−	0.934387i	$-0.615948\pi$
$431$	−14.7922	−0.712518	−0.356259	+	0.934387i	$0.615948\pi$
$432$	−3.69202	−0.177632
$433$	−16.5321	−0.794483	−0.397242	−	0.917714i	$-0.630032\pi$
$433$	−16.5321	−0.794483	−0.397242	+	0.917714i	$0.630032\pi$
$434$	9.42758	0.452538
$435$	0	0
$436$	−6.09783	−0.292033
$437$	3.08575	0.147612
$438$	−12.9554	−0.619033
$439$	3.50125	0.167106	0.0835529	−	0.996503i	$-0.473373\pi$
$439$	3.50125	0.167106	0.0835529	+	0.996503i	$0.473373\pi$
$440$	0	0
$441$	−6.91053	−0.329073
$442$	0	0
$443$	−17.4077	−0.827066	−0.413533	−	0.910489i	$-0.635705\pi$
$443$	−17.4077	−0.827066	−0.413533	+	0.910489i	$0.635705\pi$
$444$	10.0194	0.475499
$445$	0	0
$446$	−12.9879	−0.614996
$447$	−38.2935	−1.81122
$448$	1.10992	0.0524386
$449$	34.1497	1.61163	0.805813	−	0.592170i	$-0.201729\pi$
$449$	34.1497	1.61163	0.805813	+	0.592170i	$0.201729\pi$
$450$	0	0
$451$	16.9366	0.797514
$452$	12.2010	0.573889
$453$	0.650874	0.0305807
$454$	13.8049	0.647897
$455$	0	0
$456$	1.86831	0.0874918
$457$	−9.40342	−0.439873	−0.219937	−	0.975514i	$-0.570585\pi$
$457$	−9.40342	−0.439873	−0.219937	+	0.975514i	$0.570585\pi$
$458$	−11.5603	−0.540179
$459$	22.0073	1.02721
$460$	0	0
$461$	−0.733169	−0.0341471	−0.0170735	−	0.999854i	$-0.505435\pi$
$461$	−0.733169	−0.0341471	−0.0170735	+	0.999854i	$0.505435\pi$
$462$	−5.35988	−0.249364
$463$	−7.24267	−0.336595	−0.168298	−	0.985736i	$-0.553827\pi$
$463$	−7.24267	−0.336595	−0.168298	+	0.985736i	$0.553827\pi$
$464$	−3.78017	−0.175490
$465$	0	0
$466$	9.77479	0.452808
$467$	−30.2446	−1.39955	−0.699776	−	0.714362i	$-0.746717\pi$
$467$	−30.2446	−1.39955	−0.699776	+	0.714362i	$0.746717\pi$
$468$	0	0
$469$	2.38271	0.110024
$470$	0	0
$471$	−38.7042	−1.78340
$472$	10.1468	0.467042
$473$	1.21552	0.0558897
$474$	31.7211	1.45700
$475$	0	0
$476$	−6.61596	−0.303242
$477$	4.05429	0.185633
$478$	−0.944378	−0.0431948
$479$	−36.7198	−1.67777	−0.838884	−	0.544310i	$-0.816792\pi$
$479$	−36.7198	−1.67777	−0.838884	+	0.544310i	$0.816792\pi$
$480$	0	0
$481$	0	0
$482$	−0.219833	−0.0100131
$483$	−7.69574	−0.350168
$484$	−5.44504	−0.247502
$485$	0	0
$486$	11.7875	0.534690
$487$	28.6547	1.29847	0.649234	−	0.760588i	$-0.275089\pi$
$487$	28.6547	1.29847	0.649234	+	0.760588i	$0.275089\pi$
$488$	0.439665	0.0199027
$489$	−8.87741	−0.401450
$490$	0	0
$491$	30.4295	1.37326	0.686632	−	0.727005i	$-0.259088\pi$
$491$	30.4295	1.37326	0.686632	+	0.727005i	$0.259088\pi$
$492$	14.7235	0.663786
$493$	22.5327	1.01482
$494$	0	0
$495$	0	0
$496$	−8.49396	−0.381390
$497$	−0.683661	−0.0306664
$498$	1.86831	0.0837211
$499$	15.9715	0.714984	0.357492	−	0.933916i	$-0.383632\pi$
$499$	15.9715	0.714984	0.357492	+	0.933916i	$0.383632\pi$
$500$	0	0
$501$	28.6848	1.28154
$502$	−16.2543	−0.725464
$503$	41.9711	1.87140	0.935698	−	0.352801i	$-0.114771\pi$
$503$	41.9711	1.87140	0.935698	+	0.352801i	$0.114771\pi$
$504$	−1.32975	−0.0592317
$505$	0	0
$506$	7.97584	0.354569
$507$	0	0
$508$	−11.4276	−0.507017
$509$	0.914247	0.0405233	0.0202616	−	0.999795i	$-0.493550\pi$
$509$	0.914247	0.0405233	0.0202616	+	0.999795i	$0.493550\pi$
$510$	0	0
$511$	7.01805	0.310460
$512$	−1.00000	−0.0441942
$513$	3.36658	0.148638
$514$	−22.4373	−0.989666
$515$	0	0
$516$	1.05669	0.0465181
$517$	16.4698	0.724341
$518$	−5.42758	−0.238474
$519$	22.5133	0.988226
$520$	0	0
$521$	−3.31096	−0.145056	−0.0725279	−	0.997366i	$-0.523107\pi$
$521$	−3.31096	−0.145056	−0.0725279	+	0.997366i	$0.523107\pi$
$522$	4.52888	0.198224
$523$	0.850855	0.0372053	0.0186026	−	0.999827i	$-0.494078\pi$
$523$	0.850855	0.0372053	0.0186026	+	0.999827i	$0.494078\pi$
$524$	2.29590	0.100297
$525$	0	0
$526$	−10.4940	−0.457558
$527$	50.6305	2.20550
$528$	4.82908	0.210159
$529$	−11.5483	−0.502098
$530$	0	0
$531$	−12.1564	−0.527545
$532$	−1.01208	−0.0438793
$533$	0	0
$534$	7.68473	0.332551
$535$	0	0
$536$	−2.14675	−0.0927256
$537$	9.53809	0.411599
$538$	−26.4155	−1.13885
$539$	−13.5948	−0.585569
$540$	0	0
$541$	−40.8853	−1.75780	−0.878898	−	0.477010i	$-0.841721\pi$
$541$	−40.8853	−1.75780	−0.878898	+	0.477010i	$0.841721\pi$
$542$	−22.0301	−0.946275
$543$	−2.18492	−0.0937637
$544$	5.96077	0.255566
$545$	0	0
$546$	0	0
$547$	2.39075	0.102221	0.0511105	−	0.998693i	$-0.483724\pi$
$547$	2.39075	0.102221	0.0511105	+	0.998693i	$0.483724\pi$
$548$	9.08038	0.387894
$549$	−0.526746	−0.0224810
$550$	0	0
$551$	3.44696	0.146845
$552$	6.93362	0.295115
$553$	−17.1836	−0.730720
$554$	2.17629	0.0924618
$555$	0	0
$556$	18.9051	0.801757
$557$	−27.1508	−1.15042	−0.575208	−	0.818007i	$-0.695079\pi$
$557$	−27.1508	−1.15042	−0.575208	+	0.818007i	$0.695079\pi$
$558$	10.1763	0.430797
$559$	0	0
$560$	0	0
$561$	−28.7851	−1.21531
$562$	25.0030	1.05469
$563$	6.52409	0.274958	0.137479	−	0.990505i	$-0.456100\pi$
$563$	6.52409	0.274958	0.137479	+	0.990505i	$0.456100\pi$
$564$	14.3177	0.602883
$565$	0	0
$566$	−16.3153	−0.685782
$567$	−12.3854	−0.520137
$568$	0.615957	0.0258450
$569$	7.30021	0.306041	0.153020	−	0.988223i	$-0.451100\pi$
$569$	7.30021	0.306041	0.153020	+	0.988223i	$0.451100\pi$
$570$	0	0
$571$	−43.6722	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$571$	−43.6722	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$572$	0	0
$573$	1.82371	0.0761865
$574$	−7.97584	−0.332905
$575$	0	0
$576$	1.19806	0.0499193
$577$	−16.8528	−0.701590	−0.350795	−	0.936452i	$-0.614089\pi$
$577$	−16.8528	−0.701590	−0.350795	+	0.936452i	$0.614089\pi$
$578$	−18.5308	−0.770779
$579$	33.2282	1.38092
$580$	0	0
$581$	−1.01208	−0.0419882
$582$	30.0670	1.24632
$583$	7.97584	0.330325
$584$	−6.32304	−0.261649
$585$	0	0
$586$	1.87800	0.0775796
$587$	22.1825	0.915571	0.457785	−	0.889063i	$-0.348643\pi$
$587$	22.1825	0.915571	0.457785	+	0.889063i	$0.348643\pi$
$588$	−11.8183	−0.487380
$589$	7.74525	0.319137
$590$	0	0
$591$	−23.5034	−0.966800
$592$	4.89008	0.200981
$593$	−3.98493	−0.163642	−0.0818208	−	0.996647i	$-0.526074\pi$
$593$	−3.98493	−0.163642	−0.0818208	+	0.996647i	$0.526074\pi$
$594$	8.70171	0.357035
$595$	0	0
$596$	−18.6896	−0.765557
$597$	−7.78495	−0.318617
$598$	0	0
$599$	33.2379	1.35806	0.679032	−	0.734109i	$-0.262400\pi$
$599$	33.2379	1.35806	0.679032	+	0.734109i	$0.262400\pi$
$600$	0	0
$601$	9.79715	0.399634	0.199817	−	0.979833i	$-0.435965\pi$
$601$	9.79715	0.399634	0.199817	+	0.979833i	$0.435965\pi$
$602$	−0.572417	−0.0233300
$603$	2.57194	0.104738
$604$	0.317667	0.0129257
$605$	0	0
$606$	−17.9651	−0.729782
$607$	24.2258	0.983295	0.491647	−	0.870794i	$-0.336395\pi$
$607$	24.2258	0.983295	0.491647	+	0.870794i	$0.336395\pi$
$608$	0.911854	0.0369806
$609$	−8.59658	−0.348351
$610$	0	0
$611$	0	0
$612$	−7.14138	−0.288673
$613$	15.0556	0.608091	0.304045	−	0.952658i	$-0.401663\pi$
$613$	15.0556	0.608091	0.304045	+	0.952658i	$0.401663\pi$
$614$	23.9801	0.967760
$615$	0	0
$616$	−2.61596	−0.105400
$617$	2.01879	0.0812733	0.0406366	−	0.999174i	$-0.487061\pi$
$617$	2.01879	0.0812733	0.0406366	+	0.999174i	$0.487061\pi$
$618$	38.5435	1.55045
$619$	−7.84309	−0.315240	−0.157620	−	0.987500i	$-0.550382\pi$
$619$	−7.84309	−0.315240	−0.157620	+	0.987500i	$0.550382\pi$
$620$	0	0
$621$	12.4940	0.501365
$622$	5.38404	0.215880
$623$	−4.16288	−0.166782
$624$	0	0
$625$	0	0
$626$	18.9487	0.757342
$627$	−4.40342	−0.175856
$628$	−18.8901	−0.753796
$629$	−29.1487	−1.16223
$630$	0	0
$631$	24.5327	0.976632	0.488316	−	0.872667i	$-0.337611\pi$
$631$	24.5327	0.976632	0.488316	+	0.872667i	$0.337611\pi$
$632$	15.4819	0.615836
$633$	−51.3183	−2.03972
$634$	11.5013	0.456773
$635$	0	0
$636$	6.93362	0.274936
$637$	0	0
$638$	8.90946	0.352729
$639$	−0.737955	−0.0291930
$640$	0	0
$641$	−41.6015	−1.64316	−0.821580	−	0.570093i	$-0.806907\pi$
$641$	−41.6015	−1.64316	−0.821580	+	0.570093i	$0.806907\pi$
$642$	36.9869	1.45975
$643$	−45.4118	−1.79087	−0.895433	−	0.445196i	$-0.853134\pi$
$643$	−45.4118	−1.79087	−0.895433	+	0.445196i	$0.853134\pi$
$644$	−3.75600	−0.148007
$645$	0	0
$646$	−5.43535	−0.213851
$647$	−35.8345	−1.40880	−0.704399	−	0.709804i	$-0.748783\pi$
$647$	−35.8345	−1.40880	−0.704399	+	0.709804i	$0.748783\pi$
$648$	11.1588	0.438360
$649$	−23.9148	−0.938740
$650$	0	0
$651$	−19.3163	−0.757067
$652$	−4.33273	−0.169683
$653$	−18.5590	−0.726270	−0.363135	−	0.931737i	$-0.618294\pi$
$653$	−18.5590	−0.726270	−0.363135	+	0.931737i	$0.618294\pi$
$654$	12.4940	0.488552
$655$	0	0
$656$	7.18598	0.280565
$657$	7.57540	0.295545
$658$	−7.75600	−0.302361
$659$	−3.97525	−0.154854	−0.0774268	−	0.996998i	$-0.524670\pi$
$659$	−3.97525	−0.154854	−0.0774268	+	0.996998i	$0.524670\pi$
$660$	0	0
$661$	1.23191	0.0479159	0.0239580	−	0.999713i	$-0.492373\pi$
$661$	1.23191	0.0479159	0.0239580	+	0.999713i	$0.492373\pi$
$662$	−34.6112	−1.34520
$663$	0	0
$664$	0.911854	0.0353868
$665$	0	0
$666$	−5.85862	−0.227017
$667$	12.7922	0.495318
$668$	14.0000	0.541676
$669$	26.6112	1.02885
$670$	0	0
$671$	−1.03624	−0.0400038
$672$	−2.27413	−0.0877263
$673$	36.8256	1.41952	0.709762	−	0.704442i	$-0.248803\pi$
$673$	36.8256	1.41952	0.709762	+	0.704442i	$0.248803\pi$
$674$	1.95407	0.0752678
$675$	0	0
$676$	0	0
$677$	25.9215	0.996246	0.498123	−	0.867106i	$-0.334023\pi$
$677$	25.9215	0.996246	0.498123	+	0.867106i	$0.334023\pi$
$678$	−24.9989	−0.960078
$679$	−16.2875	−0.625058
$680$	0	0
$681$	−28.2851	−1.08389
$682$	20.0194	0.766582
$683$	−37.5472	−1.43670	−0.718352	−	0.695680i	$-0.755103\pi$
$683$	−37.5472	−1.43670	−0.718352	+	0.695680i	$0.755103\pi$
$684$	−1.09246	−0.0417712
$685$	0	0
$686$	14.1715	0.541071
$687$	23.6862	0.903684
$688$	0.515729	0.0196620
$689$	0	0
$690$	0	0
$691$	45.2549	1.72158	0.860788	−	0.508963i	$-0.169971\pi$
$691$	45.2549	1.72158	0.860788	+	0.508963i	$0.169971\pi$
$692$	10.9879	0.417698
$693$	3.13408	0.119054
$694$	−6.41550	−0.243529
$695$	0	0
$696$	7.74525	0.293583
$697$	−42.8340	−1.62245
$698$	−1.08575	−0.0410964
$699$	−20.0277	−0.757519
$700$	0	0
$701$	36.0823	1.36281	0.681405	−	0.731907i	$-0.261369\pi$
$701$	36.0823	1.36281	0.681405	+	0.731907i	$0.261369\pi$
$702$	0	0
$703$	−4.45904	−0.168176
$704$	2.35690	0.0888289
$705$	0	0
$706$	4.28919	0.161426
$707$	9.73184	0.366004
$708$	−20.7899	−0.781331
$709$	−19.0664	−0.716053	−0.358026	−	0.933711i	$-0.616550\pi$
$709$	−19.0664	−0.716053	−0.358026	+	0.933711i	$0.616550\pi$
$710$	0	0
$711$	−18.5483	−0.695614
$712$	3.75063	0.140561
$713$	28.7439	1.07647
$714$	13.5555	0.507304
$715$	0	0
$716$	4.65519	0.173972
$717$	1.93495	0.0722621
$718$	15.5060	0.578680
$719$	−15.3056	−0.570802	−0.285401	−	0.958408i	$-0.592127\pi$
$719$	−15.3056	−0.570802	−0.285401	+	0.958408i	$0.592127\pi$
$720$	0	0
$721$	−20.8793	−0.777587
$722$	18.1685	0.676162
$723$	0.450419	0.0167513
$724$	−1.06638	−0.0396315
$725$	0	0
$726$	11.1564	0.414054
$727$	−3.46250	−0.128417	−0.0642085	−	0.997937i	$-0.520452\pi$
$727$	−3.46250	−0.128417	−0.0642085	+	0.997937i	$0.520452\pi$
$728$	0	0
$729$	9.32496	0.345369
$730$	0	0
$731$	−3.07415	−0.113701
$732$	−0.900837	−0.0332959
$733$	26.0930	0.963769	0.481884	−	0.876235i	$-0.339953\pi$
$733$	26.0930	0.963769	0.481884	+	0.876235i	$0.339953\pi$
$734$	17.4276	0.643264
$735$	0	0
$736$	3.38404	0.124737
$737$	5.05967	0.186375
$738$	−8.60925	−0.316911
$739$	−26.4993	−0.974794	−0.487397	−	0.873181i	$-0.662054\pi$
$739$	−26.4993	−0.974794	−0.487397	+	0.873181i	$0.662054\pi$
$740$	0	0
$741$	0	0
$742$	−3.75600	−0.137887
$743$	0.415502	0.0152433	0.00762164	−	0.999971i	$-0.497574\pi$
$743$	0.415502	0.0152433	0.00762164	+	0.999971i	$0.497574\pi$
$744$	17.4034	0.638040
$745$	0	0
$746$	8.19567	0.300065
$747$	−1.09246	−0.0399709
$748$	−14.0489	−0.513679
$749$	−20.0361	−0.732103
$750$	0	0
$751$	2.90946	0.106168	0.0530839	−	0.998590i	$-0.483095\pi$
$751$	2.90946	0.106168	0.0530839	+	0.998590i	$0.483095\pi$
$752$	6.98792	0.254823
$753$	33.3037	1.21365
$754$	0	0
$755$	0	0
$756$	−4.09783	−0.149037
$757$	−12.3720	−0.449667	−0.224833	−	0.974397i	$-0.572184\pi$
$757$	−12.3720	−0.449667	−0.224833	+	0.974397i	$0.572184\pi$
$758$	15.0476	0.546553
$759$	−16.3418	−0.593171
$760$	0	0
$761$	42.4306	1.53811	0.769053	−	0.639184i	$-0.220728\pi$
$761$	42.4306	1.53811	0.769053	+	0.639184i	$0.220728\pi$
$762$	23.4142	0.848206
$763$	−6.76809	−0.245021
$764$	0.890084	0.0322021
$765$	0	0
$766$	−11.1207	−0.401806
$767$	0	0
$768$	2.04892	0.0739339
$769$	13.3341	0.480839	0.240419	−	0.970669i	$-0.422715\pi$
$769$	13.3341	0.480839	0.240419	+	0.970669i	$0.422715\pi$
$770$	0	0
$771$	45.9721	1.65565
$772$	16.2174	0.583678
$773$	5.85384	0.210548	0.105274	−	0.994443i	$-0.466428\pi$
$773$	5.85384	0.210548	0.105274	+	0.994443i	$0.466428\pi$
$774$	−0.617876	−0.0222091
$775$	0	0
$776$	14.6746	0.526786
$777$	11.1207	0.398952
$778$	8.04354	0.288375
$779$	−6.55257	−0.234770
$780$	0	0
$781$	−1.45175	−0.0519476
$782$	−20.1715	−0.721332
$783$	13.9565	0.498763
$784$	−5.76809	−0.206003
$785$	0	0
$786$	−4.70410	−0.167790
$787$	23.2965	0.830430	0.415215	−	0.909723i	$-0.363706\pi$
$787$	23.2965	0.830430	0.415215	+	0.909723i	$0.363706\pi$
$788$	−11.4711	−0.408642
$789$	21.5013	0.765465
$790$	0	0
$791$	13.5421	0.481503
$792$	−2.82371	−0.100336
$793$	0	0
$794$	−21.9081	−0.777491
$795$	0	0
$796$	−3.79954	−0.134671
$797$	−35.8103	−1.26847	−0.634233	−	0.773142i	$-0.718684\pi$
$797$	−35.8103	−1.26847	−0.634233	+	0.773142i	$0.718684\pi$
$798$	2.07367	0.0734072
$799$	−41.6534	−1.47359
$800$	0	0
$801$	−4.49349	−0.158769
$802$	−17.4426	−0.615921
$803$	14.9028	0.525907
$804$	4.39852	0.155124
$805$	0	0
$806$	0	0
$807$	54.1232	1.90523
$808$	−8.76809	−0.308460
$809$	28.3744	0.997589	0.498795	−	0.866720i	$-0.333776\pi$
$809$	28.3744	0.997589	0.498795	+	0.866720i	$0.333776\pi$
$810$	0	0
$811$	5.20344	0.182717	0.0913587	−	0.995818i	$-0.470879\pi$
$811$	5.20344	0.182717	0.0913587	+	0.995818i	$0.470879\pi$
$812$	−4.19567	−0.147239
$813$	45.1379	1.58306
$814$	−11.5254	−0.403966
$815$	0	0
$816$	−12.2131	−0.427545
$817$	−0.470270	−0.0164527
$818$	17.4330	0.609529
$819$	0	0
$820$	0	0
$821$	−5.65338	−0.197304	−0.0986522	−	0.995122i	$-0.531453\pi$
$821$	−5.65338	−0.197304	−0.0986522	+	0.995122i	$0.531453\pi$
$822$	−18.6049	−0.648922
$823$	−39.0616	−1.36160	−0.680801	−	0.732469i	$-0.738368\pi$
$823$	−39.0616	−1.36160	−0.680801	+	0.732469i	$0.738368\pi$
$824$	18.8116	0.655334
$825$	0	0
$826$	11.2620	0.391857
$827$	5.40283	0.187875	0.0939374	−	0.995578i	$-0.470055\pi$
$827$	5.40283	0.187875	0.0939374	+	0.995578i	$0.470055\pi$
$828$	−4.05429	−0.140896
$829$	−8.38537	−0.291236	−0.145618	−	0.989341i	$-0.546517\pi$
$829$	−8.38537	−0.291236	−0.145618	+	0.989341i	$0.546517\pi$
$830$	0	0
$831$	−4.45904	−0.154682
$832$	0	0
$833$	34.3822	1.19127
$834$	−38.7351	−1.34129
$835$	0	0
$836$	−2.14914	−0.0743297
$837$	31.3599	1.08396
$838$	−9.97584	−0.344610
$839$	−3.98062	−0.137426	−0.0687132	−	0.997636i	$-0.521889\pi$
$839$	−3.98062	−0.137426	−0.0687132	+	0.997636i	$0.521889\pi$
$840$	0	0
$841$	−14.7103	−0.507253
$842$	−0.615957	−0.0212273
$843$	−51.2290	−1.76442
$844$	−25.0465	−0.862137
$845$	0	0
$846$	−8.37196	−0.287834
$847$	−6.04354	−0.207659
$848$	3.38404	0.116209
$849$	33.4286	1.14727
$850$	0	0
$851$	−16.5483	−0.567267
$852$	−1.26205	−0.0432370
$853$	−6.29350	−0.215485	−0.107743	−	0.994179i	$-0.534362\pi$
$853$	−6.29350	−0.215485	−0.107743	+	0.994179i	$0.534362\pi$
$854$	0.487991	0.0166987
$855$	0	0
$856$	18.0519	0.617001
$857$	−4.37627	−0.149491	−0.0747453	−	0.997203i	$-0.523814\pi$
$857$	−4.37627	−0.149491	−0.0747453	+	0.997203i	$0.523814\pi$
$858$	0	0
$859$	15.0261	0.512683	0.256342	−	0.966586i	$-0.417483\pi$
$859$	15.0261	0.512683	0.256342	+	0.966586i	$0.417483\pi$
$860$	0	0
$861$	16.3418	0.556928
$862$	14.7922	0.503826
$863$	−6.21121	−0.211432	−0.105716	−	0.994396i	$-0.533713\pi$
$863$	−6.21121	−0.211432	−0.105716	+	0.994396i	$0.533713\pi$
$864$	3.69202	0.125605
$865$	0	0
$866$	16.5321	0.561784
$867$	37.9681	1.28946
$868$	−9.42758	−0.319993
$869$	−36.4892	−1.23781
$870$	0	0
$871$	0	0
$872$	6.09783	0.206499
$873$	−17.5810	−0.595028
$874$	−3.08575	−0.104377
$875$	0	0
$876$	12.9554	0.437722
$877$	−38.2198	−1.29059	−0.645296	−	0.763933i	$-0.723266\pi$
$877$	−38.2198	−1.29059	−0.645296	+	0.763933i	$0.723266\pi$
$878$	−3.50125	−0.118162
$879$	−3.84787	−0.129785
$880$	0	0
$881$	26.7832	0.902347	0.451174	−	0.892436i	$-0.351006\pi$
$881$	26.7832	0.902347	0.451174	+	0.892436i	$0.351006\pi$
$882$	6.91053	0.232690
$883$	34.4956	1.16087	0.580435	−	0.814307i	$-0.302883\pi$
$883$	34.4956	1.16087	0.580435	+	0.814307i	$0.302883\pi$
$884$	0	0
$885$	0	0
$886$	17.4077	0.584824
$887$	23.9866	0.805391	0.402695	−	0.915334i	$-0.368073\pi$
$887$	23.9866	0.805391	0.402695	+	0.915334i	$0.368073\pi$
$888$	−10.0194	−0.336228
$889$	−12.6837	−0.425396
$890$	0	0
$891$	−26.3002	−0.881090
$892$	12.9879	0.434868
$893$	−6.37196	−0.213230
$894$	38.2935	1.28073
$895$	0	0
$896$	−1.10992	−0.0370797
$897$	0	0
$898$	−34.1497	−1.13959
$899$	32.1086	1.07088
$900$	0	0
$901$	−20.1715	−0.672010
$902$	−16.9366	−0.563927
$903$	1.17283	0.0390295
$904$	−12.2010	−0.405801
$905$	0	0
$906$	−0.650874	−0.0216238
$907$	−23.9269	−0.794480	−0.397240	−	0.917715i	$-0.630032\pi$
$907$	−23.9269	−0.794480	−0.397240	+	0.917715i	$0.630032\pi$
$908$	−13.8049	−0.458132
$909$	10.5047	0.348419
$910$	0	0
$911$	−35.8866	−1.18898	−0.594488	−	0.804104i	$-0.702645\pi$
$911$	−35.8866	−1.18898	−0.594488	+	0.804104i	$0.702645\pi$
$912$	−1.86831	−0.0618660
$913$	−2.14914	−0.0711263
$914$	9.40342	0.311037
$915$	0	0
$916$	11.5603	0.381964
$917$	2.54825	0.0841507
$918$	−22.0073	−0.726349
$919$	33.2465	1.09670	0.548351	−	0.836249i	$-0.315256\pi$
$919$	33.2465	1.09670	0.548351	+	0.836249i	$0.315256\pi$
$920$	0	0
$921$	−49.1333	−1.61900
$922$	0.733169	0.0241456
$923$	0	0
$924$	5.35988	0.176327
$925$	0	0
$926$	7.24267	0.238009
$927$	−22.5375	−0.740229
$928$	3.78017	0.124090
$929$	54.2583	1.78016	0.890079	−	0.455806i	$-0.150649\pi$
$929$	54.2583	1.78016	0.890079	+	0.455806i	$0.150649\pi$
$930$	0	0
$931$	5.25965	0.172378
$932$	−9.77479	−0.320184
$933$	−11.0315	−0.361154
$934$	30.2446	0.989633
$935$	0	0
$936$	0	0
$937$	−16.5265	−0.539897	−0.269948	−	0.962875i	$-0.587007\pi$
$937$	−16.5265	−0.539897	−0.269948	+	0.962875i	$0.587007\pi$
$938$	−2.38271	−0.0777984
$939$	−38.8243	−1.26698
$940$	0	0
$941$	41.7017	1.35944	0.679718	−	0.733473i	$-0.262102\pi$
$941$	41.7017	1.35944	0.679718	+	0.733473i	$0.262102\pi$
$942$	38.7042	1.26105
$943$	−24.3177	−0.791892
$944$	−10.1468	−0.330249
$945$	0	0
$946$	−1.21552	−0.0395200
$947$	3.00106	0.0975215	0.0487608	−	0.998810i	$-0.484473\pi$
$947$	3.00106	0.0975215	0.0487608	+	0.998810i	$0.484473\pi$
$948$	−31.7211	−1.03025
$949$	0	0
$950$	0	0
$951$	−23.5651	−0.764151
$952$	6.61596	0.214424
$953$	−38.1450	−1.23564	−0.617818	−	0.786321i	$-0.711983\pi$
$953$	−38.1450	−1.23564	−0.617818	+	0.786321i	$0.711983\pi$
$954$	−4.05429	−0.131263
$955$	0	0
$956$	0.944378	0.0305434
$957$	−18.2547	−0.590092
$958$	36.7198	1.18636
$959$	10.0785	0.325450
$960$	0	0
$961$	41.1473	1.32733
$962$	0	0
$963$	−21.6273	−0.696930
$964$	0.219833	0.00708033
$965$	0	0
$966$	7.69574	0.247606
$967$	−26.8793	−0.864381	−0.432190	−	0.901782i	$-0.642259\pi$
$967$	−26.8793	−0.864381	−0.432190	+	0.901782i	$0.642259\pi$
$968$	5.44504	0.175010
$969$	11.1366	0.357759
$970$	0	0
$971$	3.13647	0.100654	0.0503271	−	0.998733i	$-0.483974\pi$
$971$	3.13647	0.100654	0.0503271	+	0.998733i	$0.483974\pi$
$972$	−11.7875	−0.378083
$973$	20.9831	0.672688
$974$	−28.6547	−0.918156
$975$	0	0
$976$	−0.439665	−0.0140733
$977$	35.8864	1.14811	0.574053	−	0.818818i	$-0.305370\pi$
$977$	35.8864	1.14811	0.574053	+	0.818818i	$0.305370\pi$
$978$	8.87741	0.283868
$979$	−8.83984	−0.282522
$980$	0	0
$981$	−7.30559	−0.233249
$982$	−30.4295	−0.971044
$983$	−30.4370	−0.970790	−0.485395	−	0.874295i	$-0.661324\pi$
$983$	−30.4370	−0.970790	−0.485395	+	0.874295i	$0.661324\pi$
$984$	−14.7235	−0.469367
$985$	0	0
$986$	−22.5327	−0.717588
$987$	15.8914	0.505829
$988$	0	0
$989$	−1.74525	−0.0554957
$990$	0	0
$991$	31.4470	0.998946	0.499473	−	0.866330i	$-0.333527\pi$
$991$	31.4470	0.998946	0.499473	+	0.866330i	$0.333527\pi$
$992$	8.49396	0.269683
$993$	70.9154	2.25043
$994$	0.683661	0.0216844
$995$	0	0
$996$	−1.86831	−0.0591998
$997$	−19.1099	−0.605217	−0.302609	−	0.953115i	$-0.597857\pi$
$997$	−19.1099	−0.605217	−0.302609	+	0.953115i	$0.597857\pi$
$998$	−15.9715	−0.505570
$999$	−18.0543	−0.571213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bn.1.3		3
5.4	even	2		338.2.a.h.1.1	yes	3
13.12	even	2		8450.2.a.bx.1.3		3
15.14	odd	2		3042.2.a.z.1.1		3
20.19	odd	2		2704.2.a.w.1.3		3
65.4	even	6		338.2.c.i.315.3		6
65.9	even	6		338.2.c.h.315.3		6
65.19	odd	12		338.2.e.e.23.3		12
65.24	odd	12		338.2.e.e.147.3		12
65.29	even	6		338.2.c.h.191.3		6
65.34	odd	4		338.2.b.d.337.4		6
65.44	odd	4		338.2.b.d.337.1		6
65.49	even	6		338.2.c.i.191.3		6
65.54	odd	12		338.2.e.e.147.6		12
65.59	odd	12		338.2.e.e.23.6		12
65.64	even	2		338.2.a.g.1.1	✓	3
195.44	even	4		3042.2.b.n.1351.6		6
195.164	even	4		3042.2.b.n.1351.1		6
195.194	odd	2		3042.2.a.bi.1.3		3
260.99	even	4		2704.2.f.m.337.6		6
260.239	even	4		2704.2.f.m.337.5		6
260.259	odd	2		2704.2.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.1	✓	3	65.64	even	2
338.2.a.h.1.1	yes	3	5.4	even	2
338.2.b.d.337.1		6	65.44	odd	4
338.2.b.d.337.4		6	65.34	odd	4
338.2.c.h.191.3		6	65.29	even	6
338.2.c.h.315.3		6	65.9	even	6
338.2.c.i.191.3		6	65.49	even	6
338.2.c.i.315.3		6	65.4	even	6
338.2.e.e.23.3		12	65.19	odd	12
338.2.e.e.23.6		12	65.59	odd	12
338.2.e.e.147.3		12	65.24	odd	12
338.2.e.e.147.6		12	65.54	odd	12
2704.2.a.v.1.3		3	260.259	odd	2
2704.2.a.w.1.3		3	20.19	odd	2
2704.2.f.m.337.5		6	260.239	even	4
2704.2.f.m.337.6		6	260.99	even	4
3042.2.a.z.1.1		3	15.14	odd	2
3042.2.a.bi.1.3		3	195.194	odd	2
3042.2.b.n.1351.1		6	195.164	even	4
3042.2.b.n.1351.6		6	195.44	even	4
8450.2.a.bn.1.3		3	1.1	even	1	trivial
8450.2.a.bx.1.3		3	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.04892 q^{3} +1.00000 q^{4} -2.04892 q^{6} +1.10992 q^{7} -1.00000 q^{8} +1.19806 q^{9} +2.35690 q^{11} +2.04892 q^{12} -1.10992 q^{14} +1.00000 q^{16} -5.96077 q^{17} -1.19806 q^{18} -0.911854 q^{19} +2.27413 q^{21} -2.35690 q^{22} -3.38404 q^{23} -2.04892 q^{24} -3.69202 q^{27} +1.10992 q^{28} -3.78017 q^{29} -8.49396 q^{31} -1.00000 q^{32} +4.82908 q^{33} +5.96077 q^{34} +1.19806 q^{36} +4.89008 q^{37} +0.911854 q^{38} +7.18598 q^{41} -2.27413 q^{42} +0.515729 q^{43} +2.35690 q^{44} +3.38404 q^{46} +6.98792 q^{47} +2.04892 q^{48} -5.76809 q^{49} -12.2131 q^{51} +3.38404 q^{53} +3.69202 q^{54} -1.10992 q^{56} -1.86831 q^{57} +3.78017 q^{58} -10.1468 q^{59} -0.439665 q^{61} +8.49396 q^{62} +1.32975 q^{63} +1.00000 q^{64} -4.82908 q^{66} +2.14675 q^{67} -5.96077 q^{68} -6.93362 q^{69} -0.615957 q^{71} -1.19806 q^{72} +6.32304 q^{73} -4.89008 q^{74} -0.911854 q^{76} +2.61596 q^{77} -15.4819 q^{79} -11.1588 q^{81} -7.18598 q^{82} -0.911854 q^{83} +2.27413 q^{84} -0.515729 q^{86} -7.74525 q^{87} -2.35690 q^{88} -3.75063 q^{89} -3.38404 q^{92} -17.4034 q^{93} -6.98792 q^{94} -2.04892 q^{96} -14.6746 q^{97} +5.76809 q^{98} +2.82371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{11} - 3 q^{12} - 4 q^{14} + 3 q^{16} - 5 q^{17} - 8 q^{18} + q^{19} - 4 q^{21} - 3 q^{22} + 3 q^{24} - 6 q^{27} + 4 q^{28}+ \cdots + q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^11 - 3 * q^12 - 4 * q^14 + 3 * q^16 - 5 * q^17 - 8 * q^18 + q^19 - 4 * q^21 - 3 * q^22 + 3 * q^24 - 6 * q^27 + 4 * q^28 - 10 * q^29 - 16 * q^31 - 3 * q^32 + 4 * q^33 + 5 * q^34 + 8 * q^36 + 14 * q^37 - q^38 + 7 * q^41 + 4 * q^42 - 11 * q^43 + 3 * q^44 + 2 * q^47 - 3 * q^48 + 3 * q^49 - 16 * q^51 + 6 * q^54 - 4 * q^56 - 8 * q^57 + 10 * q^58 - 3 * q^59 - 4 * q^61 + 16 * q^62 + 6 * q^63 + 3 * q^64 - 4 * q^66 - 21 * q^67 - 5 * q^68 - 14 * q^69 - 12 * q^71 - 8 * q^72 - q^73 - 14 * q^74 + q^76 + 18 * q^77 - 18 * q^79 - 25 * q^81 - 7 * q^82 + q^83 - 4 * q^84 + 11 * q^86 + 10 * q^87 - 3 * q^88 + 25 * q^89 + 2 * q^93 - 2 * q^94 + 3 * q^96 - 23 * q^97 - 3 * q^98 + q^99

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