LMFDB - Embedded newform 8450.2.a.bq.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,-3,0,3,0,0,-3,-3,9,0,-3,0,0,3,0,3,9] 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:	3.3.2808.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 2 $ x^3 - 9*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 650)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.10548$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +3.10548 q^{3} +1.00000 q^{4} -3.10548 q^{6} -4.10548 q^{7} -1.00000 q^{8} +6.64402 q^{9} -1.53854 q^{11} +3.10548 q^{12} +4.10548 q^{14} +1.00000 q^{16} +3.00000 q^{17} -6.64402 q^{18} +1.10548 q^{19} -12.7495 q^{21} +1.53854 q^{22} -6.74950 q^{23} -3.10548 q^{24} +11.3164 q^{27} -4.10548 q^{28} -5.56694 q^{29} -8.64402 q^{31} -1.00000 q^{32} -4.77791 q^{33} -3.00000 q^{34} +6.64402 q^{36} +4.53854 q^{37} -1.10548 q^{38} +7.85499 q^{41} +12.7495 q^{42} +4.00000 q^{43} -1.53854 q^{44} +6.74950 q^{46} +10.8550 q^{47} +3.10548 q^{48} +9.85499 q^{49} +9.31645 q^{51} +0.433057 q^{53} -11.3164 q^{54} +4.10548 q^{56} +3.43306 q^{57} +5.56694 q^{58} -13.7779 q^{59} -14.3164 q^{61} +8.64402 q^{62} -27.2769 q^{63} +1.00000 q^{64} +4.77791 q^{66} -3.74950 q^{67} +3.00000 q^{68} -20.9605 q^{69} -6.64402 q^{72} +10.7779 q^{73} -4.53854 q^{74} +1.10548 q^{76} +6.31645 q^{77} -6.53854 q^{79} +15.2110 q^{81} -7.85499 q^{82} -4.46146 q^{83} -12.7495 q^{84} -4.00000 q^{86} -17.2880 q^{87} +1.53854 q^{88} -1.85499 q^{89} -6.74950 q^{92} -26.8439 q^{93} -10.8550 q^{94} -3.10548 q^{96} -3.78903 q^{97} -9.85499 q^{98} -10.2221 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{16} + 9 q^{17} - 9 q^{18} - 6 q^{19} - 18 q^{21} + 3 q^{22} + 6 q^{27} - 3 q^{28} - 9 q^{29} - 15 q^{31} - 3 q^{32} + 12 q^{33}+ \cdots - 57 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^9 - 3 * q^11 + 3 * q^14 + 3 * q^16 + 9 * q^17 - 9 * q^18 - 6 * q^19 - 18 * q^21 + 3 * q^22 + 6 * q^27 - 3 * q^28 - 9 * q^29 - 15 * q^31 - 3 * q^32 + 12 * q^33 - 9 * q^34 + 9 * q^36 + 12 * q^37 + 6 * q^38 - 6 * q^41 + 18 * q^42 + 12 * q^43 - 3 * q^44 + 3 * q^47 + 9 * q^53 - 6 * q^54 + 3 * q^56 + 18 * q^57 + 9 * q^58 - 15 * q^59 - 15 * q^61 + 15 * q^62 - 15 * q^63 + 3 * q^64 - 12 * q^66 + 9 * q^67 + 9 * q^68 - 24 * q^69 - 9 * q^72 + 6 * q^73 - 12 * q^74 - 6 * q^76 - 9 * q^77 - 18 * q^79 + 27 * q^81 + 6 * q^82 - 15 * q^83 - 18 * q^84 - 12 * q^86 - 30 * q^87 + 3 * q^88 + 24 * q^89 - 6 * q^93 - 3 * q^94 - 30 * q^97 - 57 * 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
$3$	3.10548	1.79295	0.896476	−	0.443093i	$-0.146119\pi$
$3$	3.10548	1.79295	0.896476	+	0.443093i	$0.146119\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.10548	−1.26781
$7$	−4.10548	−1.55173	−0.775863	−	0.630901i	$-0.782685\pi$
$7$	−4.10548	−1.55173	−0.775863	+	0.630901i	$0.782685\pi$
$8$	−1.00000	−0.353553
$9$	6.64402	2.21467
$10$	0	0
$11$	−1.53854	−0.463887	−0.231944	−	0.972729i	$-0.574508\pi$
$11$	−1.53854	−0.463887	−0.231944	+	0.972729i	$0.574508\pi$
$12$	3.10548	0.896476
$13$	0	0
$13$	0	0
$14$	4.10548	1.09724
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−6.64402	−1.56601
$19$	1.10548	0.253615	0.126808	−	0.991927i	$-0.459527\pi$
$19$	1.10548	0.253615	0.126808	+	0.991927i	$0.459527\pi$
$20$	0	0
$21$	−12.7495	−2.78217
$22$	1.53854	0.328018
$23$	−6.74950	−1.40737	−0.703685	−	0.710513i	$-0.748463\pi$
$23$	−6.74950	−1.40737	−0.703685	+	0.710513i	$0.748463\pi$
$24$	−3.10548	−0.633904
$25$	0	0
$26$	0	0
$27$	11.3164	2.17785
$28$	−4.10548	−0.775863
$29$	−5.56694	−1.03376	−0.516878	−	0.856059i	$-0.672906\pi$
$29$	−5.56694	−1.03376	−0.516878	+	0.856059i	$0.672906\pi$
$30$	0	0
$31$	−8.64402	−1.55251	−0.776256	−	0.630418i	$-0.782884\pi$
$31$	−8.64402	−1.55251	−0.776256	+	0.630418i	$0.782884\pi$
$32$	−1.00000	−0.176777
$33$	−4.77791	−0.831727
$34$	−3.00000	−0.514496
$35$	0	0
$36$	6.64402	1.10734
$37$	4.53854	0.746131	0.373066	−	0.927805i	$-0.378307\pi$
$37$	4.53854	0.746131	0.373066	+	0.927805i	$0.378307\pi$
$38$	−1.10548	−0.179333
$39$	0	0
$40$	0	0
$41$	7.85499	1.22674	0.613371	−	0.789795i	$-0.289813\pi$
$41$	7.85499	1.22674	0.613371	+	0.789795i	$0.289813\pi$
$42$	12.7495	1.96729
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.53854	−0.231944
$45$	0	0
$46$	6.74950	0.995160
$47$	10.8550	1.58336	0.791681	−	0.610934i	$-0.209206\pi$
$47$	10.8550	1.58336	0.791681	+	0.610934i	$0.209206\pi$
$48$	3.10548	0.448238
$49$	9.85499	1.40786
$50$	0	0
$51$	9.31645	1.30456
$52$	0	0
$53$	0.433057	0.0594850	0.0297425	−	0.999558i	$-0.490531\pi$
$53$	0.433057	0.0594850	0.0297425	+	0.999558i	$0.490531\pi$
$54$	−11.3164	−1.53997
$55$	0	0
$56$	4.10548	0.548618
$57$	3.43306	0.454720
$58$	5.56694	0.730975
$59$	−13.7779	−1.79373	−0.896865	−	0.442304i	$-0.854161\pi$
$59$	−13.7779	−1.79373	−0.896865	+	0.442304i	$0.854161\pi$
$60$	0	0
$61$	−14.3164	−1.83303	−0.916517	−	0.399997i	$-0.869011\pi$
$61$	−14.3164	−1.83303	−0.916517	+	0.399997i	$0.869011\pi$
$62$	8.64402	1.09779
$63$	−27.2769	−3.43657
$64$	1.00000	0.125000
$65$	0	0
$66$	4.77791	0.588120
$67$	−3.74950	−0.458075	−0.229037	−	0.973418i	$-0.573558\pi$
$67$	−3.74950	−0.458075	−0.229037	+	0.973418i	$0.573558\pi$
$68$	3.00000	0.363803
$69$	−20.9605	−2.52334
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−6.64402	−0.783006
$73$	10.7779	1.26146	0.630729	−	0.776003i	$-0.282756\pi$
$73$	10.7779	1.26146	0.630729	+	0.776003i	$0.282756\pi$
$74$	−4.53854	−0.527595
$75$	0	0
$76$	1.10548	0.126808
$77$	6.31645	0.719826
$78$	0	0
$79$	−6.53854	−0.735643	−0.367822	−	0.929896i	$-0.619896\pi$
$79$	−6.53854	−0.735643	−0.367822	+	0.929896i	$0.619896\pi$
$80$	0	0
$81$	15.2110	1.69011
$82$	−7.85499	−0.867438
$83$	−4.46146	−0.489709	−0.244854	−	0.969560i	$-0.578740\pi$
$83$	−4.46146	−0.489709	−0.244854	+	0.969560i	$0.578740\pi$
$84$	−12.7495	−1.39109
$85$	0	0
$86$	−4.00000	−0.431331
$87$	−17.2880	−1.85347
$88$	1.53854	0.164009
$89$	−1.85499	−0.196628	−0.0983141	−	0.995155i	$-0.531345\pi$
$89$	−1.85499	−0.196628	−0.0983141	+	0.995155i	$0.531345\pi$
$90$	0	0
$91$	0	0
$92$	−6.74950	−0.703685
$93$	−26.8439	−2.78358
$94$	−10.8550	−1.11961
$95$	0	0
$96$	−3.10548	−0.316952
$97$	−3.78903	−0.384718	−0.192359	−	0.981325i	$-0.561614\pi$
$97$	−3.78903	−0.384718	−0.192359	+	0.981325i	$0.561614\pi$
$98$	−9.85499	−0.995504
$99$	−10.2221	−1.02736
$100$	0	0
$101$	−7.18256	−0.714692	−0.357346	−	0.933972i	$-0.616318\pi$
$101$	−7.18256	−0.714692	−0.357346	+	0.933972i	$0.616318\pi$
$102$	−9.31645	−0.922466
$103$	−12.5385	−1.23546	−0.617730	−	0.786391i	$-0.711947\pi$
$103$	−12.5385	−1.23546	−0.617730	+	0.786391i	$0.711947\pi$
$104$	0	0
$105$	0	0
$106$	−0.433057	−0.0420622
$107$	1.81744	0.175698	0.0878492	−	0.996134i	$-0.472001\pi$
$107$	1.81744	0.175698	0.0878492	+	0.996134i	$0.472001\pi$
$108$	11.3164	1.08893
$109$	−3.67243	−0.351755	−0.175877	−	0.984412i	$-0.556276\pi$
$109$	−3.67243	−0.351755	−0.175877	+	0.984412i	$0.556276\pi$
$110$	0	0
$111$	14.0944	1.33778
$112$	−4.10548	−0.387932
$113$	−16.0660	−1.51136	−0.755679	−	0.654942i	$-0.772693\pi$
$113$	−16.0660	−1.51136	−0.755679	+	0.654942i	$0.772693\pi$
$114$	−3.43306	−0.321535
$115$	0	0
$116$	−5.56694	−0.516878
$117$	0	0
$118$	13.7779	1.26836
$119$	−12.3164	−1.12905
$120$	0	0
$121$	−8.63290	−0.784809
$122$	14.3164	1.29615
$123$	24.3935	2.19949
$124$	−8.64402	−0.776256
$125$	0	0
$126$	27.2769	2.43002
$127$	4.92292	0.436839	0.218419	−	0.975855i	$-0.429910\pi$
$127$	4.92292	0.436839	0.218419	+	0.975855i	$0.429910\pi$
$128$	−1.00000	−0.0883883
$129$	12.4219	1.09369
$130$	0	0
$131$	0.866114	0.0756727	0.0378364	−	0.999284i	$-0.487953\pi$
$131$	0.866114	0.0756727	0.0378364	+	0.999284i	$0.487953\pi$
$132$	−4.77791	−0.415864
$133$	−4.53854	−0.393541
$134$	3.74950	0.323908
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−16.7779	−1.43343	−0.716717	−	0.697364i	$-0.754356\pi$
$137$	−16.7779	−1.43343	−0.716717	+	0.697364i	$0.754356\pi$
$138$	20.9605	1.78427
$139$	−4.39353	−0.372654	−0.186327	−	0.982488i	$-0.559658\pi$
$139$	−4.39353	−0.372654	−0.186327	+	0.982488i	$0.559658\pi$
$140$	0	0
$141$	33.7100	2.83889
$142$	0	0
$143$	0	0
$144$	6.64402	0.553669
$145$	0	0
$146$	−10.7779	−0.891986
$147$	30.6045	2.52422
$148$	4.53854	0.373066
$149$	9.70998	0.795472	0.397736	−	0.917500i	$-0.369796\pi$
$149$	9.70998	0.795472	0.397736	+	0.917500i	$0.369796\pi$
$150$	0	0
$151$	−2.48986	−0.202622	−0.101311	−	0.994855i	$-0.532304\pi$
$151$	−2.48986	−0.202622	−0.101311	+	0.994855i	$0.532304\pi$
$152$	−1.10548	−0.0896665
$153$	19.9321	1.61141
$154$	−6.31645	−0.508994
$155$	0	0
$156$	0	0
$157$	11.3935	0.909302	0.454651	−	0.890670i	$-0.349764\pi$
$157$	11.3935	0.909302	0.454651	+	0.890670i	$0.349764\pi$
$158$	6.53854	0.520178
$159$	1.34485	0.106654
$160$	0	0
$161$	27.7100	2.18385
$162$	−15.2110	−1.19509
$163$	−16.8155	−1.31709	−0.658544	−	0.752542i	$-0.728827\pi$
$163$	−16.8155	−1.31709	−0.658544	+	0.752542i	$0.728827\pi$
$164$	7.85499	0.613371
$165$	0	0
$166$	4.46146	0.346276
$167$	−1.61562	−0.125020	−0.0625102	−	0.998044i	$-0.519911\pi$
$167$	−1.61562	−0.125020	−0.0625102	+	0.998044i	$0.519911\pi$
$168$	12.7495	0.983646
$169$	0	0
$170$	0	0
$171$	7.34485	0.561675
$172$	4.00000	0.304997
$173$	−12.3164	−0.936402	−0.468201	−	0.883622i	$-0.655098\pi$
$173$	−12.3164	−0.936402	−0.468201	+	0.883622i	$0.655098\pi$
$174$	17.2880	1.31060
$175$	0	0
$176$	−1.53854	−0.115972
$177$	−42.7871	−3.21607
$178$	1.85499	0.139037
$179$	−10.1826	−0.761080	−0.380540	−	0.924764i	$-0.624262\pi$
$179$	−10.1826	−0.761080	−0.380540	+	0.924764i	$0.624262\pi$
$180$	0	0
$181$	−11.3935	−0.846874	−0.423437	−	0.905926i	$-0.639177\pi$
$181$	−11.3935	−0.846874	−0.423437	+	0.905926i	$0.639177\pi$
$182$	0	0
$183$	−44.4595	−3.28654
$184$	6.74950	0.497580
$185$	0	0
$186$	26.8439	1.96829
$187$	−4.61562	−0.337527
$188$	10.8550	0.791681
$189$	−46.4595	−3.37943
$190$	0	0
$191$	−19.4990	−1.41090	−0.705449	−	0.708760i	$-0.749255\pi$
$191$	−19.4990	−1.41090	−0.705449	+	0.708760i	$0.749255\pi$
$192$	3.10548	0.224119
$193$	2.72110	0.195869	0.0979346	−	0.995193i	$-0.468776\pi$
$193$	2.72110	0.195869	0.0979346	+	0.995193i	$0.468776\pi$
$194$	3.78903	0.272037
$195$	0	0
$196$	9.85499	0.703928
$197$	1.61562	0.115108	0.0575540	−	0.998342i	$-0.481670\pi$
$197$	1.61562	0.115108	0.0575540	+	0.998342i	$0.481670\pi$
$198$	10.2221	0.726452
$199$	22.2485	1.57716	0.788578	−	0.614935i	$-0.210818\pi$
$199$	22.2485	1.57716	0.788578	+	0.614935i	$0.210818\pi$
$200$	0	0
$201$	−11.6440	−0.821306
$202$	7.18256	0.505363
$203$	22.8550	1.60411
$204$	9.31645	0.652282
$205$	0	0
$206$	12.5385	0.873601
$207$	−44.8439	−3.11686
$208$	0	0
$209$	−1.70083	−0.117649
$210$	0	0
$211$	10.2394	0.704907	0.352454	−	0.935829i	$-0.385347\pi$
$211$	10.2394	0.704907	0.352454	+	0.935829i	$0.385347\pi$
$212$	0.433057	0.0297425
$213$	0	0
$214$	−1.81744	−0.124238
$215$	0	0
$216$	−11.3164	−0.769987
$217$	35.4879	2.40907
$218$	3.67243	0.248728
$219$	33.4706	2.26173
$220$	0	0
$221$	0	0
$222$	−14.0944	−0.945951
$223$	−20.9605	−1.40362	−0.701808	−	0.712366i	$-0.747624\pi$
$223$	−20.9605	−1.40362	−0.701808	+	0.712366i	$0.747624\pi$
$224$	4.10548	0.274309
$225$	0	0
$226$	16.0660	1.06869
$227$	4.85499	0.322237	0.161118	−	0.986935i	$-0.448490\pi$
$227$	4.85499	0.322237	0.161118	+	0.986935i	$0.448490\pi$
$228$	3.43306	0.227360
$229$	−6.71196	−0.443538	−0.221769	−	0.975099i	$-0.571183\pi$
$229$	−6.71196	−0.443538	−0.221769	+	0.975099i	$0.571183\pi$
$230$	0	0
$231$	19.6156	1.29061
$232$	5.56694	0.365488
$233$	14.3651	0.941091	0.470545	−	0.882376i	$-0.344057\pi$
$233$	14.3651	0.941091	0.470545	+	0.882376i	$0.344057\pi$
$234$	0	0
$235$	0	0
$236$	−13.7779	−0.896865
$237$	−20.3053	−1.31897
$238$	12.3164	0.798357
$239$	1.14501	0.0740647	0.0370324	−	0.999314i	$-0.488210\pi$
$239$	1.14501	0.0740647	0.0370324	+	0.999314i	$0.488210\pi$
$240$	0	0
$241$	−20.5669	−1.32483	−0.662417	−	0.749136i	$-0.730469\pi$
$241$	−20.5669	−1.32483	−0.662417	+	0.749136i	$0.730469\pi$
$242$	8.63290	0.554944
$243$	13.2880	0.852428
$244$	−14.3164	−0.916517
$245$	0	0
$246$	−24.3935	−1.55527
$247$	0	0
$248$	8.64402	0.548896
$249$	−13.8550	−0.878024
$250$	0	0
$251$	−3.31645	−0.209332	−0.104666	−	0.994507i	$-0.533377\pi$
$251$	−3.31645	−0.209332	−0.104666	+	0.994507i	$0.533377\pi$
$252$	−27.2769	−1.71828
$253$	10.3844	0.652860
$254$	−4.92292	−0.308892
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.4331	0.775553	0.387776	−	0.921753i	$-0.373243\pi$
$257$	12.4331	0.775553	0.387776	+	0.921753i	$0.373243\pi$
$258$	−12.4219	−0.773356
$259$	−18.6329	−1.15779
$260$	0	0
$261$	−36.9869	−2.28943
$262$	−0.866114	−0.0535087
$263$	−4.38438	−0.270353	−0.135176	−	0.990822i	$-0.543160\pi$
$263$	−4.38438	−0.270353	−0.135176	+	0.990822i	$0.543160\pi$
$264$	4.77791	0.294060
$265$	0	0
$266$	4.53854	0.278276
$267$	−5.76063	−0.352545
$268$	−3.74950	−0.229037
$269$	−5.56694	−0.339423	−0.169711	−	0.985494i	$-0.554284\pi$
$269$	−5.56694	−0.339423	−0.169711	+	0.985494i	$0.554284\pi$
$270$	0	0
$271$	9.39353	0.570616	0.285308	−	0.958436i	$-0.407904\pi$
$271$	9.39353	0.570616	0.285308	+	0.958436i	$0.407904\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	16.7779	1.01359
$275$	0	0
$276$	−20.9605	−1.26167
$277$	8.38438	0.503769	0.251884	−	0.967757i	$-0.418950\pi$
$277$	8.38438	0.503769	0.251884	+	0.967757i	$0.418950\pi$
$278$	4.39353	0.263506
$279$	−57.4311	−3.43831
$280$	0	0
$281$	12.6329	0.753615	0.376808	−	0.926292i	$-0.377022\pi$
$281$	12.6329	0.753615	0.376808	+	0.926292i	$0.377022\pi$
$282$	−33.7100	−2.00740
$283$	−13.4706	−0.800744	−0.400372	−	0.916353i	$-0.631119\pi$
$283$	−13.4706	−0.800744	−0.400372	+	0.916353i	$0.631119\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.2485	−1.90357
$288$	−6.64402	−0.391503
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−11.7668	−0.689781
$292$	10.7779	0.630729
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−30.6045	−1.78489
$295$	0	0
$296$	−4.53854	−0.263797
$297$	−17.4108	−1.01028
$298$	−9.70998	−0.562484
$299$	0	0
$300$	0	0
$301$	−16.4219	−0.946544
$302$	2.48986	0.143276
$303$	−22.3053	−1.28141
$304$	1.10548	0.0634038
$305$	0	0
$306$	−19.9321	−1.13944
$307$	13.8925	0.792889	0.396444	−	0.918059i	$-0.370244\pi$
$307$	13.8925	0.792889	0.396444	+	0.918059i	$0.370244\pi$
$308$	6.31645	0.359913
$309$	−38.9382	−2.21512
$310$	0	0
$311$	−3.51827	−0.199503	−0.0997513	−	0.995012i	$-0.531805\pi$
$311$	−3.51827	−0.199503	−0.0997513	+	0.995012i	$0.531805\pi$
$312$	0	0
$313$	24.5650	1.38849	0.694247	−	0.719737i	$-0.255738\pi$
$313$	24.5650	1.38849	0.694247	+	0.719737i	$0.255738\pi$
$314$	−11.3935	−0.642974
$315$	0	0
$316$	−6.53854	−0.367822
$317$	−1.46146	−0.0820838	−0.0410419	−	0.999157i	$-0.513068\pi$
$317$	−1.46146	−0.0820838	−0.0410419	+	0.999157i	$0.513068\pi$
$318$	−1.34485	−0.0754155
$319$	8.56496	0.479546
$320$	0	0
$321$	5.64402	0.315019
$322$	−27.7100	−1.54422
$323$	3.31645	0.184532
$324$	15.2110	0.845054
$325$	0	0
$326$	16.8155	0.931322
$327$	−11.4047	−0.630679
$328$	−7.85499	−0.433719
$329$	−44.5650	−2.45695
$330$	0	0
$331$	−18.2394	−1.00253	−0.501263	−	0.865295i	$-0.667131\pi$
$331$	−18.2394	−1.00253	−0.501263	+	0.865295i	$0.667131\pi$
$332$	−4.46146	−0.244854
$333$	30.1542	1.65244
$334$	1.61562	0.0884027
$335$	0	0
$336$	−12.7495	−0.695543
$337$	−19.6329	−1.06947	−0.534736	−	0.845019i	$-0.679589\pi$
$337$	−19.6329	−1.06947	−0.534736	+	0.845019i	$0.679589\pi$
$338$	0	0
$339$	−49.8925	−2.70979
$340$	0	0
$341$	13.2992	0.720190
$342$	−7.34485	−0.397164
$343$	−11.7211	−0.632880
$344$	−4.00000	−0.215666
$345$	0	0
$346$	12.3164	0.662136
$347$	23.6816	1.27129	0.635647	−	0.771980i	$-0.280734\pi$
$347$	23.6816	1.27129	0.635647	+	0.771980i	$0.280734\pi$
$348$	−17.2880	−0.926736
$349$	−0.154159	−0.00825192	−0.00412596	−	0.999991i	$-0.501313\pi$
$349$	−0.154159	−0.00825192	−0.00412596	+	0.999991i	$0.501313\pi$
$350$	0	0
$351$	0	0
$352$	1.53854	0.0820044
$353$	36.6329	1.94977	0.974886	−	0.222704i	$-0.0714883\pi$
$353$	36.6329	1.94977	0.974886	+	0.222704i	$0.0714883\pi$
$354$	42.7871	2.27411
$355$	0	0
$356$	−1.85499	−0.0983141
$357$	−38.2485	−2.02433
$358$	10.1826	0.538165
$359$	−27.2394	−1.43764	−0.718819	−	0.695197i	$-0.755317\pi$
$359$	−27.2394	−1.43764	−0.718819	+	0.695197i	$0.755317\pi$
$360$	0	0
$361$	−17.7779	−0.935679
$362$	11.3935	0.598830
$363$	−26.8093	−1.40712
$364$	0	0
$365$	0	0
$366$	44.4595	2.32393
$367$	23.8641	1.24570	0.622849	−	0.782342i	$-0.285975\pi$
$367$	23.8641	1.24570	0.622849	+	0.782342i	$0.285975\pi$
$368$	−6.74950	−0.351842
$369$	52.1887	2.71684
$370$	0	0
$371$	−1.77791	−0.0923044
$372$	−26.8439	−1.39179
$373$	22.3164	1.15550	0.577751	−	0.816213i	$-0.303930\pi$
$373$	22.3164	1.15550	0.577751	+	0.816213i	$0.303930\pi$
$374$	4.61562	0.238668
$375$	0	0
$376$	−10.8550	−0.559803
$377$	0	0
$378$	46.4595	2.38962
$379$	9.74950	0.500798	0.250399	−	0.968143i	$-0.419438\pi$
$379$	9.74950	0.500798	0.250399	+	0.968143i	$0.419438\pi$
$380$	0	0
$381$	15.2880	0.783230
$382$	19.4990	0.997656
$383$	−24.7871	−1.26656	−0.633280	−	0.773923i	$-0.718292\pi$
$383$	−24.7871	−1.26656	−0.633280	+	0.773923i	$0.718292\pi$
$384$	−3.10548	−0.158476
$385$	0	0
$386$	−2.72110	−0.138500
$387$	26.5761	1.35094
$388$	−3.78903	−0.192359
$389$	0.749505	0.0380014	0.0190007	−	0.999819i	$-0.493952\pi$
$389$	0.749505	0.0380014	0.0190007	+	0.999819i	$0.493952\pi$
$390$	0	0
$391$	−20.2485	−1.02401
$392$	−9.85499	−0.497752
$393$	2.68970	0.135678
$394$	−1.61562	−0.0813937
$395$	0	0
$396$	−10.2221	−0.513679
$397$	−9.67243	−0.485445	−0.242723	−	0.970096i	$-0.578040\pi$
$397$	−9.67243	−0.485445	−0.242723	+	0.970096i	$0.578040\pi$
$398$	−22.2485	−1.11522
$399$	−14.0944	−0.705600
$400$	0	0
$401$	−10.9321	−0.545921	−0.272961	−	0.962025i	$-0.588003\pi$
$401$	−10.9321	−0.545921	−0.272961	+	0.962025i	$0.588003\pi$
$402$	11.6440	0.580751
$403$	0	0
$404$	−7.18256	−0.357346
$405$	0	0
$406$	−22.8550	−1.13427
$407$	−6.98272	−0.346121
$408$	−9.31645	−0.461233
$409$	−26.5669	−1.31365	−0.656825	−	0.754043i	$-0.728101\pi$
$409$	−26.5669	−1.31365	−0.656825	+	0.754043i	$0.728101\pi$
$410$	0	0
$411$	−52.1035	−2.57008
$412$	−12.5385	−0.617730
$413$	56.5650	2.78338
$414$	44.8439	2.20396
$415$	0	0
$416$	0	0
$417$	−13.6440	−0.668151
$418$	1.70083	0.0831903
$419$	24.3145	1.18784	0.593920	−	0.804524i	$-0.297580\pi$
$419$	24.3145	1.18784	0.593920	+	0.804524i	$0.297580\pi$
$420$	0	0
$421$	−6.15416	−0.299935	−0.149968	−	0.988691i	$-0.547917\pi$
$421$	−6.15416	−0.299935	−0.149968	+	0.988691i	$0.547917\pi$
$422$	−10.2394	−0.498445
$423$	72.1208	3.50663
$424$	−0.433057	−0.0210311
$425$	0	0
$426$	0	0
$427$	58.7759	2.84437
$428$	1.81744	0.0878492
$429$	0	0
$430$	0	0
$431$	0.828565	0.0399106	0.0199553	−	0.999801i	$-0.493648\pi$
$431$	0.828565	0.0399106	0.0199553	+	0.999801i	$0.493648\pi$
$432$	11.3164	0.544463
$433$	15.7008	0.754534	0.377267	−	0.926105i	$-0.376864\pi$
$433$	15.7008	0.754534	0.377267	+	0.926105i	$0.376864\pi$
$434$	−35.4879	−1.70347
$435$	0	0
$436$	−3.67243	−0.175877
$437$	−7.46146	−0.356930
$438$	−33.4706	−1.59929
$439$	9.17144	0.437729	0.218864	−	0.975755i	$-0.429765\pi$
$439$	9.17144	0.437729	0.218864	+	0.975755i	$0.429765\pi$
$440$	0	0
$441$	65.4768	3.11794
$442$	0	0
$443$	−15.3164	−0.727706	−0.363853	−	0.931456i	$-0.618539\pi$
$443$	−15.3164	−0.727706	−0.363853	+	0.931456i	$0.618539\pi$
$444$	14.0944	0.668889
$445$	0	0
$446$	20.9605	0.992507
$447$	30.1542	1.42624
$448$	−4.10548	−0.193966
$449$	23.4108	1.10482	0.552412	−	0.833571i	$-0.313708\pi$
$449$	23.4108	1.10482	0.552412	+	0.833571i	$0.313708\pi$
$450$	0	0
$451$	−12.0852	−0.569070
$452$	−16.0660	−0.755679
$453$	−7.73223	−0.363292
$454$	−4.85499	−0.227856
$455$	0	0
$456$	−3.43306	−0.160768
$457$	22.9321	1.07272	0.536358	−	0.843990i	$-0.319800\pi$
$457$	22.9321	1.07272	0.536358	+	0.843990i	$0.319800\pi$
$458$	6.71196	0.313629
$459$	33.9493	1.58462
$460$	0	0
$461$	−13.4615	−0.626963	−0.313481	−	0.949594i	$-0.601495\pi$
$461$	−13.4615	−0.626963	−0.313481	+	0.949594i	$0.601495\pi$
$462$	−19.6156	−0.912601
$463$	−16.0264	−0.744811	−0.372406	−	0.928070i	$-0.621467\pi$
$463$	−16.0264	−0.744811	−0.372406	+	0.928070i	$0.621467\pi$
$464$	−5.56694	−0.258439
$465$	0	0
$466$	−14.3651	−0.665452
$467$	−0.866114	−0.0400790	−0.0200395	−	0.999799i	$-0.506379\pi$
$467$	−0.866114	−0.0400790	−0.0200395	+	0.999799i	$0.506379\pi$
$468$	0	0
$469$	15.3935	0.710807
$470$	0	0
$471$	35.3824	1.63033
$472$	13.7779	0.634180
$473$	−6.15416	−0.282969
$474$	20.3053	0.932654
$475$	0	0
$476$	−12.3164	−0.564523
$477$	2.87724	0.131740
$478$	−1.14501	−0.0523717
$479$	−29.4879	−1.34734	−0.673668	−	0.739034i	$-0.735282\pi$
$479$	−29.4879	−1.34734	−0.673668	+	0.739034i	$0.735282\pi$
$480$	0	0
$481$	0	0
$482$	20.5669	0.936799
$483$	86.0528	3.91554
$484$	−8.63290	−0.392404
$485$	0	0
$486$	−13.2880	−0.602758
$487$	−35.4503	−1.60641	−0.803204	−	0.595704i	$-0.796873\pi$
$487$	−35.4503	−1.60641	−0.803204	+	0.595704i	$0.796873\pi$
$488$	14.3164	0.648075
$489$	−52.2201	−2.36148
$490$	0	0
$491$	32.1319	1.45009	0.725046	−	0.688700i	$-0.241818\pi$
$491$	32.1319	1.45009	0.725046	+	0.688700i	$0.241818\pi$
$492$	24.3935	1.09975
$493$	−16.7008	−0.752168
$494$	0	0
$495$	0	0
$496$	−8.64402	−0.388128
$497$	0	0
$498$	13.8550	0.620857
$499$	−27.2769	−1.22108	−0.610541	−	0.791984i	$-0.709048\pi$
$499$	−27.2769	−1.22108	−0.610541	+	0.791984i	$0.709048\pi$
$500$	0	0
$501$	−5.01728	−0.224155
$502$	3.31645	0.148020
$503$	−26.9980	−1.20378	−0.601891	−	0.798578i	$-0.705586\pi$
$503$	−26.9980	−1.20378	−0.601891	+	0.798578i	$0.705586\pi$
$504$	27.2769	1.21501
$505$	0	0
$506$	−10.3844	−0.461642
$507$	0	0
$508$	4.92292	0.218419
$509$	38.8814	1.72339	0.861694	−	0.507428i	$-0.169404\pi$
$509$	38.8814	1.72339	0.861694	+	0.507428i	$0.169404\pi$
$510$	0	0
$511$	−44.2485	−1.95744
$512$	−1.00000	−0.0441942
$513$	12.5101	0.552336
$514$	−12.4331	−0.548399
$515$	0	0
$516$	12.4219	0.546845
$517$	−16.7008	−0.734502
$518$	18.6329	0.818682
$519$	−38.2485	−1.67892
$520$	0	0
$521$	4.06595	0.178133	0.0890663	−	0.996026i	$-0.471612\pi$
$521$	4.06595	0.178133	0.0890663	+	0.996026i	$0.471612\pi$
$522$	36.9869	1.61887
$523$	−34.8723	−1.52486	−0.762429	−	0.647072i	$-0.775993\pi$
$523$	−34.8723	−1.52486	−0.762429	+	0.647072i	$0.775993\pi$
$524$	0.866114	0.0378364
$525$	0	0
$526$	4.38438	0.191168
$527$	−25.9321	−1.12962
$528$	−4.77791	−0.207932
$529$	22.5558	0.980688
$530$	0	0
$531$	−91.5407	−3.97253
$532$	−4.53854	−0.196771
$533$	0	0
$534$	5.76063	0.249287
$535$	0	0
$536$	3.74950	0.161954
$537$	−31.6218	−1.36458
$538$	5.56694	0.240008
$539$	−15.1623	−0.653086
$540$	0	0
$541$	37.2282	1.60057	0.800284	−	0.599622i	$-0.204682\pi$
$541$	37.2282	1.60057	0.800284	+	0.599622i	$0.204682\pi$
$542$	−9.39353	−0.403487
$543$	−35.3824	−1.51840
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	35.9493	1.53708	0.768541	−	0.639800i	$-0.220983\pi$
$547$	35.9493	1.53708	0.768541	+	0.639800i	$0.220983\pi$
$548$	−16.7779	−0.716717
$549$	−95.1188	−4.05957
$550$	0	0
$551$	−6.15416	−0.262176
$552$	20.9605	0.892137
$553$	26.8439	1.14152
$554$	−8.38438	−0.356218
$555$	0	0
$556$	−4.39353	−0.186327
$557$	14.0944	0.597197	0.298599	−	0.954379i	$-0.403481\pi$
$557$	14.0944	0.597197	0.298599	+	0.954379i	$0.403481\pi$
$558$	57.4311	2.43125
$559$	0	0
$560$	0	0
$561$	−14.3337	−0.605170
$562$	−12.6329	−0.532887
$563$	−28.2678	−1.19134	−0.595672	−	0.803228i	$-0.703114\pi$
$563$	−28.2678	−1.19134	−0.595672	+	0.803228i	$0.703114\pi$
$564$	33.7100	1.41945
$565$	0	0
$566$	13.4706	0.566212
$567$	−62.4484	−2.62258
$568$	0	0
$569$	26.1339	1.09559	0.547795	−	0.836613i	$-0.315467\pi$
$569$	26.1339	1.09559	0.547795	+	0.836613i	$0.315467\pi$
$570$	0	0
$571$	−10.7871	−0.451424	−0.225712	−	0.974194i	$-0.572471\pi$
$571$	−10.7871	−0.451424	−0.225712	+	0.974194i	$0.572471\pi$
$572$	0	0
$573$	−60.5538	−2.52967
$574$	32.2485	1.34603
$575$	0	0
$576$	6.64402	0.276834
$577$	12.1228	0.504677	0.252339	−	0.967639i	$-0.418800\pi$
$577$	12.1228	0.504677	0.252339	+	0.967639i	$0.418800\pi$
$578$	8.00000	0.332756
$579$	8.45033	0.351184
$580$	0	0
$581$	18.3164	0.759894
$582$	11.7668	0.487749
$583$	−0.666275	−0.0275943
$584$	−10.7779	−0.445993
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−35.2485	−1.45486	−0.727431	−	0.686181i	$-0.759286\pi$
$587$	−35.2485	−1.45486	−0.727431	+	0.686181i	$0.759286\pi$
$588$	30.6045	1.26211
$589$	−9.55582	−0.393741
$590$	0	0
$591$	5.01728	0.206383
$592$	4.53854	0.186533
$593$	−11.0862	−0.455257	−0.227628	−	0.973748i	$-0.573097\pi$
$593$	−11.0862	−0.455257	−0.227628	+	0.973748i	$0.573097\pi$
$594$	17.4108	0.714374
$595$	0	0
$596$	9.70998	0.397736
$597$	69.0924	2.82776
$598$	0	0
$599$	31.4990	1.28701	0.643507	−	0.765440i	$-0.277479\pi$
$599$	31.4990	1.28701	0.643507	+	0.765440i	$0.277479\pi$
$600$	0	0
$601$	19.6329	0.800843	0.400421	−	0.916331i	$-0.368864\pi$
$601$	19.6329	0.800843	0.400421	+	0.916331i	$0.368864\pi$
$602$	16.4219	0.669308
$603$	−24.9118	−1.01449
$604$	−2.48986	−0.101311
$605$	0	0
$606$	22.3053	0.906092
$607$	−17.6156	−0.714996	−0.357498	−	0.933914i	$-0.616370\pi$
$607$	−17.6156	−0.714996	−0.357498	+	0.933914i	$0.616370\pi$
$608$	−1.10548	−0.0448332
$609$	70.9758	2.87608
$610$	0	0
$611$	0	0
$612$	19.9321	0.805706
$613$	25.4200	1.02670	0.513351	−	0.858179i	$-0.328404\pi$
$613$	25.4200	1.02670	0.513351	+	0.858179i	$0.328404\pi$
$614$	−13.8925	−0.560657
$615$	0	0
$616$	−6.31645	−0.254497
$617$	27.7100	1.11556	0.557781	−	0.829988i	$-0.311653\pi$
$617$	27.7100	1.11556	0.557781	+	0.829988i	$0.311653\pi$
$618$	38.9382	1.56632
$619$	−20.8439	−0.837786	−0.418893	−	0.908036i	$-0.637582\pi$
$619$	−20.8439	−0.837786	−0.418893	+	0.908036i	$0.637582\pi$
$620$	0	0
$621$	−76.3804	−3.06504
$622$	3.51827	0.141070
$623$	7.61562	0.305113
$624$	0	0
$625$	0	0
$626$	−24.5650	−0.981813
$627$	−5.28189	−0.210939
$628$	11.3935	0.454651
$629$	13.6156	0.542890
$630$	0	0
$631$	−30.7495	−1.22412	−0.612059	−	0.790812i	$-0.709659\pi$
$631$	−30.7495	−1.22412	−0.612059	+	0.790812i	$0.709659\pi$
$632$	6.53854	0.260089
$633$	31.7982	1.26386
$634$	1.46146	0.0580420
$635$	0	0
$636$	1.34485	0.0533268
$637$	0	0
$638$	−8.56496	−0.339090
$639$	0	0
$640$	0	0
$641$	−19.0660	−0.753060	−0.376530	−	0.926404i	$-0.622883\pi$
$641$	−19.0660	−0.753060	−0.376530	+	0.926404i	$0.622883\pi$
$642$	−5.64402	−0.222752
$643$	45.8641	1.80870	0.904352	−	0.426786i	$-0.140354\pi$
$643$	45.8641	1.80870	0.904352	+	0.426786i	$0.140354\pi$
$644$	27.7100	1.09193
$645$	0	0
$646$	−3.31645	−0.130484
$647$	11.0173	0.433134	0.216567	−	0.976268i	$-0.430514\pi$
$647$	11.0173	0.433134	0.216567	+	0.976268i	$0.430514\pi$
$648$	−15.2110	−0.597543
$649$	21.1979	0.832089
$650$	0	0
$651$	110.207	4.31935
$652$	−16.8155	−0.658544
$653$	14.5650	0.569971	0.284986	−	0.958532i	$-0.408011\pi$
$653$	14.5650	0.569971	0.284986	+	0.958532i	$0.408011\pi$
$654$	11.4047	0.445957
$655$	0	0
$656$	7.85499	0.306686
$657$	71.6087	2.79372
$658$	44.5650	1.73732
$659$	19.8174	0.771978	0.385989	−	0.922503i	$-0.373860\pi$
$659$	19.8174	0.771978	0.385989	+	0.922503i	$0.373860\pi$
$660$	0	0
$661$	−13.4615	−0.523590	−0.261795	−	0.965123i	$-0.584314\pi$
$661$	−13.4615	−0.523590	−0.261795	+	0.965123i	$0.584314\pi$
$662$	18.2394	0.708893
$663$	0	0
$664$	4.46146	0.173138
$665$	0	0
$666$	−30.1542	−1.16845
$667$	37.5741	1.45488
$668$	−1.61562	−0.0625102
$669$	−65.0924	−2.51662
$670$	0	0
$671$	22.0264	0.850321
$672$	12.7495	0.491823
$673$	−0.0679332	−0.00261863	−0.00130932	−	0.999999i	$-0.500417\pi$
$673$	−0.0679332	−0.00261863	−0.00130932	+	0.999999i	$0.500417\pi$
$674$	19.6329	0.756231
$675$	0	0
$676$	0	0
$677$	26.1319	1.00433	0.502165	−	0.864772i	$-0.332537\pi$
$677$	26.1319	1.00433	0.502165	+	0.864772i	$0.332537\pi$
$678$	49.8925	1.91611
$679$	15.5558	0.596977
$680$	0	0
$681$	15.0771	0.577755
$682$	−13.2992	−0.509252
$683$	17.0944	0.654097	0.327049	−	0.945007i	$-0.393946\pi$
$683$	17.0944	0.654097	0.327049	+	0.945007i	$0.393946\pi$
$684$	7.34485	0.280837
$685$	0	0
$686$	11.7211	0.447514
$687$	−20.8439	−0.795243
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	−25.3844	−0.965667	−0.482834	−	0.875712i	$-0.660392\pi$
$691$	−25.3844	−0.965667	−0.482834	+	0.875712i	$0.660392\pi$
$692$	−12.3164	−0.468201
$693$	41.9666	1.59418
$694$	−23.6816	−0.898940
$695$	0	0
$696$	17.2880	0.655302
$697$	23.5650	0.892587
$698$	0.154159	0.00583499
$699$	44.6106	1.68733
$700$	0	0
$701$	−9.43108	−0.356207	−0.178103	−	0.984012i	$-0.556996\pi$
$701$	−9.43108	−0.356207	−0.178103	+	0.984012i	$0.556996\pi$
$702$	0	0
$703$	5.01728	0.189230
$704$	−1.53854	−0.0579859
$705$	0	0
$706$	−36.6329	−1.37870
$707$	29.4879	1.10901
$708$	−42.7871	−1.60804
$709$	−20.1734	−0.757629	−0.378814	−	0.925473i	$-0.623668\pi$
$709$	−20.1734	−0.757629	−0.378814	+	0.925473i	$0.623668\pi$
$710$	0	0
$711$	−43.4422	−1.62921
$712$	1.85499	0.0695186
$713$	58.3429	2.18496
$714$	38.2485	1.43141
$715$	0	0
$716$	−10.1826	−0.380540
$717$	3.55582	0.132794
$718$	27.2394	1.01656
$719$	31.4990	1.17471	0.587357	−	0.809328i	$-0.300168\pi$
$719$	31.4990	1.17471	0.587357	+	0.809328i	$0.300168\pi$
$720$	0	0
$721$	51.4768	1.91709
$722$	17.7779	0.661625
$723$	−63.8703	−2.37536
$724$	−11.3935	−0.423437
$725$	0	0
$726$	26.8093	0.994987
$727$	22.8814	0.848625	0.424312	−	0.905516i	$-0.360516\pi$
$727$	22.8814	0.848625	0.424312	+	0.905516i	$0.360516\pi$
$728$	0	0
$729$	−4.36710	−0.161745
$730$	0	0
$731$	12.0000	0.443836
$732$	−44.4595	−1.64327
$733$	48.1126	1.77708	0.888541	−	0.458798i	$-0.151720\pi$
$733$	48.1126	1.77708	0.888541	+	0.458798i	$0.151720\pi$
$734$	−23.8641	−0.880841
$735$	0	0
$736$	6.74950	0.248790
$737$	5.76876	0.212495
$738$	−52.1887	−1.92109
$739$	−19.9321	−0.733213	−0.366606	−	0.930376i	$-0.619480\pi$
$739$	−19.9321	−0.733213	−0.366606	+	0.930376i	$0.619480\pi$
$740$	0	0
$741$	0	0
$742$	1.77791	0.0652691
$743$	−24.6248	−0.903395	−0.451698	−	0.892171i	$-0.649181\pi$
$743$	−24.6248	−0.903395	−0.451698	+	0.892171i	$0.649181\pi$
$744$	26.8439	0.984144
$745$	0	0
$746$	−22.3164	−0.817063
$747$	−29.6420	−1.08455
$748$	−4.61562	−0.168764
$749$	−7.46146	−0.272636
$750$	0	0
$751$	36.7273	1.34020	0.670098	−	0.742272i	$-0.266252\pi$
$751$	36.7273	1.34020	0.670098	+	0.742272i	$0.266252\pi$
$752$	10.8550	0.395841
$753$	−10.2992	−0.375323
$754$	0	0
$755$	0	0
$756$	−46.4595	−1.68971
$757$	−37.0091	−1.34512	−0.672560	−	0.740042i	$-0.734805\pi$
$757$	−37.0091	−1.34512	−0.672560	+	0.740042i	$0.734805\pi$
$758$	−9.74950	−0.354118
$759$	32.2485	1.17055
$760$	0	0
$761$	−26.6420	−0.965773	−0.482887	−	0.875683i	$-0.660412\pi$
$761$	−26.6420	−0.965773	−0.482887	+	0.875683i	$0.660412\pi$
$762$	−15.2880	−0.553827
$763$	15.0771	0.545827
$764$	−19.4990	−0.705449
$765$	0	0
$766$	24.7871	0.895593
$767$	0	0
$768$	3.10548	0.112059
$769$	−37.2972	−1.34497	−0.672486	−	0.740110i	$-0.734773\pi$
$769$	−37.2972	−1.34497	−0.672486	+	0.740110i	$0.734773\pi$
$770$	0	0
$771$	38.6106	1.39053
$772$	2.72110	0.0979346
$773$	10.5385	0.379045	0.189522	−	0.981876i	$-0.439306\pi$
$773$	10.5385	0.379045	0.189522	+	0.981876i	$0.439306\pi$
$774$	−26.5761	−0.955258
$775$	0	0
$776$	3.78903	0.136018
$777$	−57.8641	−2.07586
$778$	−0.749505	−0.0268711
$779$	8.68355	0.311121
$780$	0	0
$781$	0	0
$782$	20.2485	0.724085
$783$	−62.9980	−2.25137
$784$	9.85499	0.351964
$785$	0	0
$786$	−2.68970	−0.0959385
$787$	−0.278898	−0.00994165	−0.00497083	−	0.999988i	$-0.501582\pi$
$787$	−0.278898	−0.00994165	−0.00497083	+	0.999988i	$0.501582\pi$
$788$	1.61562	0.0575540
$789$	−13.6156	−0.484729
$790$	0	0
$791$	65.9585	2.34521
$792$	10.2221	0.363226
$793$	0	0
$794$	9.67243	0.343262
$795$	0	0
$796$	22.2485	0.788578
$797$	−42.0832	−1.49066	−0.745332	−	0.666693i	$-0.767709\pi$
$797$	−42.0832	−1.49066	−0.745332	+	0.666693i	$0.767709\pi$
$798$	14.0944	0.498935
$799$	32.5650	1.15207
$800$	0	0
$801$	−12.3246	−0.435468
$802$	10.9321	0.386025
$803$	−16.5822	−0.585175
$804$	−11.6440	−0.410653
$805$	0	0
$806$	0	0
$807$	−17.2880	−0.608568
$808$	7.18256	0.252682
$809$	−12.6329	−0.444149	−0.222074	−	0.975030i	$-0.571283\pi$
$809$	−12.6329	−0.444149	−0.222074	+	0.975030i	$0.571283\pi$
$810$	0	0
$811$	43.1411	1.51489	0.757444	−	0.652901i	$-0.226448\pi$
$811$	43.1411	1.51489	0.757444	+	0.652901i	$0.226448\pi$
$812$	22.8550	0.802053
$813$	29.1714	1.02309
$814$	6.98272	0.244744
$815$	0	0
$816$	9.31645	0.326141
$817$	4.42193	0.154704
$818$	26.5669	0.928891
$819$	0	0
$820$	0	0
$821$	23.0173	0.803308	0.401654	−	0.915791i	$-0.368435\pi$
$821$	23.0173	0.803308	0.401654	+	0.915791i	$0.368435\pi$
$822$	52.1035	1.81732
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	12.5385	0.436801
$825$	0	0
$826$	−56.5650	−1.96815
$827$	−35.4027	−1.23107	−0.615536	−	0.788109i	$-0.711060\pi$
$827$	−35.4027	−1.23107	−0.615536	+	0.788109i	$0.711060\pi$
$828$	−44.8439	−1.55843
$829$	10.6065	0.368378	0.184189	−	0.982891i	$-0.441034\pi$
$829$	10.6065	0.368378	0.184189	+	0.982891i	$0.441034\pi$
$830$	0	0
$831$	26.0375	0.903233
$832$	0	0
$833$	29.5650	1.02437
$834$	13.6440	0.472454
$835$	0	0
$836$	−1.70083	−0.0588244
$837$	−97.8196	−3.38114
$838$	−24.3145	−0.839929
$839$	19.4615	0.671884	0.335942	−	0.941883i	$-0.390945\pi$
$839$	19.4615	0.671884	0.335942	+	0.941883i	$0.390945\pi$
$840$	0	0
$841$	1.99085	0.0686501
$842$	6.15416	0.212086
$843$	39.2312	1.35120
$844$	10.2394	0.352454
$845$	0	0
$846$	−72.1208	−2.47956
$847$	35.4422	1.21781
$848$	0.433057	0.0148712
$849$	−41.8327	−1.43570
$850$	0	0
$851$	−30.6329	−1.05008
$852$	0	0
$853$	15.9807	0.547170	0.273585	−	0.961848i	$-0.411790\pi$
$853$	15.9807	0.547170	0.273585	+	0.961848i	$0.411790\pi$
$854$	−58.7759	−2.01127
$855$	0	0
$856$	−1.81744	−0.0621188
$857$	38.5669	1.31742	0.658711	−	0.752396i	$-0.271102\pi$
$857$	38.5669	1.31742	0.658711	+	0.752396i	$0.271102\pi$
$858$	0	0
$859$	−28.6836	−0.978670	−0.489335	−	0.872096i	$-0.662760\pi$
$859$	−28.6836	−0.978670	−0.489335	+	0.872096i	$0.662760\pi$
$860$	0	0
$861$	−100.147	−3.41301
$862$	−0.828565	−0.0282210
$863$	36.4706	1.24147	0.620737	−	0.784019i	$-0.286834\pi$
$863$	36.4706	1.24147	0.620737	+	0.784019i	$0.286834\pi$
$864$	−11.3164	−0.384993
$865$	0	0
$866$	−15.7008	−0.533536
$867$	−24.8439	−0.843742
$868$	35.4879	1.20454
$869$	10.0598	0.341255
$870$	0	0
$871$	0	0
$872$	3.67243	0.124364
$873$	−25.1744	−0.852025
$874$	7.46146	0.252388
$875$	0	0
$876$	33.4706	1.13087
$877$	7.02027	0.237058	0.118529	−	0.992951i	$-0.462182\pi$
$877$	7.02027	0.237058	0.118529	+	0.992951i	$0.462182\pi$
$878$	−9.17144	−0.309521
$879$	18.6329	0.628472
$880$	0	0
$881$	−4.93405	−0.166232	−0.0831161	−	0.996540i	$-0.526487\pi$
$881$	−4.93405	−0.166232	−0.0831161	+	0.996540i	$0.526487\pi$
$882$	−65.4768	−2.20472
$883$	−49.9493	−1.68093	−0.840465	−	0.541866i	$-0.817718\pi$
$883$	−49.9493	−1.68093	−0.840465	+	0.541866i	$0.817718\pi$
$884$	0	0
$885$	0	0
$886$	15.3164	0.514566
$887$	3.86413	0.129745	0.0648725	−	0.997894i	$-0.479336\pi$
$887$	3.86413	0.129745	0.0648725	+	0.997894i	$0.479336\pi$
$888$	−14.0944	−0.472976
$889$	−20.2110	−0.677854
$890$	0	0
$891$	−23.4027	−0.784019
$892$	−20.9605	−0.701808
$893$	12.0000	0.401565
$894$	−30.1542	−1.00851
$895$	0	0
$896$	4.10548	0.137155
$897$	0	0
$898$	−23.4108	−0.781229
$899$	48.1208	1.60492
$900$	0	0
$901$	1.29917	0.0432817
$902$	12.0852	0.402393
$903$	−50.9980	−1.69711
$904$	16.0660	0.534346
$905$	0	0
$906$	7.73223	0.256886
$907$	17.5558	0.582931	0.291466	−	0.956581i	$-0.405857\pi$
$907$	17.5558	0.582931	0.291466	+	0.956581i	$0.405857\pi$
$908$	4.85499	0.161118
$909$	−47.7211	−1.58281
$910$	0	0
$911$	−27.9807	−0.927043	−0.463522	−	0.886086i	$-0.653414\pi$
$911$	−27.9807	−0.927043	−0.463522	+	0.886086i	$0.653414\pi$
$912$	3.43306	0.113680
$913$	6.86413	0.227170
$914$	−22.9321	−0.758525
$915$	0	0
$916$	−6.71196	−0.221769
$917$	−3.55582	−0.117423
$918$	−33.9493	−1.12050
$919$	31.0356	1.02377	0.511884	−	0.859054i	$-0.328948\pi$
$919$	31.0356	1.02377	0.511884	+	0.859054i	$0.328948\pi$
$920$	0	0
$921$	43.1430	1.42161
$922$	13.4615	0.443330
$923$	0	0
$924$	19.6156	0.645306
$925$	0	0
$926$	16.0264	0.526661
$927$	−83.3063	−2.73614
$928$	5.56694	0.182744
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	0	0
$931$	10.8945	0.357053
$932$	14.3651	0.470545
$933$	−10.9259	−0.357698
$934$	0.866114	0.0283401
$935$	0	0
$936$	0	0
$937$	−60.8987	−1.98947	−0.994737	−	0.102464i	$-0.967327\pi$
$937$	−60.8987	−1.98947	−0.994737	+	0.102464i	$0.967327\pi$
$938$	−15.3935	−0.502616
$939$	76.2861	2.48950
$940$	0	0
$941$	32.2485	1.05127	0.525636	−	0.850710i	$-0.323827\pi$
$941$	32.2485	1.05127	0.525636	+	0.850710i	$0.323827\pi$
$942$	−35.3824	−1.15282
$943$	−53.0173	−1.72648
$944$	−13.7779	−0.448433
$945$	0	0
$946$	6.15416	0.200089
$947$	20.4108	0.663262	0.331631	−	0.943409i	$-0.392401\pi$
$947$	20.4108	0.663262	0.331631	+	0.943409i	$0.392401\pi$
$948$	−20.3053	−0.659486
$949$	0	0
$950$	0	0
$951$	−4.53854	−0.147172
$952$	12.3164	0.399178
$953$	−8.36710	−0.271037	−0.135519	−	0.990775i	$-0.543270\pi$
$953$	−8.36710	−0.271037	−0.135519	+	0.990775i	$0.543270\pi$
$954$	−2.87724	−0.0931541
$955$	0	0
$956$	1.14501	0.0370324
$957$	26.5983	0.859802
$958$	29.4879	0.952710
$959$	68.8814	2.22430
$960$	0	0
$961$	43.7191	1.41029
$962$	0	0
$963$	12.0751	0.389115
$964$	−20.5669	−0.662417
$965$	0	0
$966$	−86.0528	−2.76870
$967$	14.4108	0.463420	0.231710	−	0.972785i	$-0.425568\pi$
$967$	14.4108	0.463420	0.231710	+	0.972785i	$0.425568\pi$
$968$	8.63290	0.277472
$969$	10.2992	0.330857
$970$	0	0
$971$	25.1806	0.808083	0.404042	−	0.914741i	$-0.367605\pi$
$971$	25.1806	0.808083	0.404042	+	0.914741i	$0.367605\pi$
$972$	13.2880	0.426214
$973$	18.0375	0.578257
$974$	35.4503	1.13590
$975$	0	0
$976$	−14.3164	−0.458258
$977$	−38.1633	−1.22095	−0.610476	−	0.792035i	$-0.709022\pi$
$977$	−38.1633	−1.22095	−0.610476	+	0.792035i	$0.709022\pi$
$978$	52.2201	1.66982
$979$	2.85397	0.0912133
$980$	0	0
$981$	−24.3997	−0.779022
$982$	−32.1319	−1.02537
$983$	49.8906	1.59126	0.795631	−	0.605782i	$-0.207140\pi$
$983$	49.8906	1.59126	0.795631	+	0.605782i	$0.207140\pi$
$984$	−24.3935	−0.777637
$985$	0	0
$986$	16.7008	0.531863
$987$	−138.396	−4.40518
$988$	0	0
$989$	−26.9980	−0.858487
$990$	0	0
$991$	−10.6329	−0.337765	−0.168883	−	0.985636i	$-0.554016\pi$
$991$	−10.6329	−0.337765	−0.168883	+	0.985636i	$0.554016\pi$
$992$	8.64402	0.274448
$993$	−56.6420	−1.79748
$994$	0	0
$995$	0	0
$996$	−13.8550	−0.439012
$997$	2.99085	0.0947213	0.0473606	−	0.998878i	$-0.484919\pi$
$997$	2.99085	0.0947213	0.0473606	+	0.998878i	$0.484919\pi$
$998$	27.2769	0.863436
$999$	51.3601	1.62496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bq.1.3		3
5.4	even	2		8450.2.a.cd.1.1		3
13.5	odd	4		650.2.d.d.51.6	yes	6
13.8	odd	4		650.2.d.d.51.3	yes	6
13.12	even	2		8450.2.a.ce.1.3		3
65.8	even	4		650.2.c.e.649.6		6
65.18	even	4		650.2.c.f.649.6		6
65.34	odd	4		650.2.d.c.51.4	yes	6
65.44	odd	4		650.2.d.c.51.1	✓	6
65.47	even	4		650.2.c.f.649.1		6
65.57	even	4		650.2.c.e.649.1		6
65.64	even	2		8450.2.a.br.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.c.e.649.1		6	65.57	even	4
650.2.c.e.649.6		6	65.8	even	4
650.2.c.f.649.1		6	65.47	even	4
650.2.c.f.649.6		6	65.18	even	4
650.2.d.c.51.1	✓	6	65.44	odd	4
650.2.d.c.51.4	yes	6	65.34	odd	4
650.2.d.d.51.3	yes	6	13.8	odd	4
650.2.d.d.51.6	yes	6	13.5	odd	4
8450.2.a.bq.1.3		3	1.1	even	1	trivial
8450.2.a.br.1.1		3	65.64	even	2
8450.2.a.cd.1.1		3	5.4	even	2
8450.2.a.ce.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.10548 q^{3} +1.00000 q^{4} -3.10548 q^{6} -4.10548 q^{7} -1.00000 q^{8} +6.64402 q^{9} -1.53854 q^{11} +3.10548 q^{12} +4.10548 q^{14} +1.00000 q^{16} +3.00000 q^{17} -6.64402 q^{18} +1.10548 q^{19} -12.7495 q^{21} +1.53854 q^{22} -6.74950 q^{23} -3.10548 q^{24} +11.3164 q^{27} -4.10548 q^{28} -5.56694 q^{29} -8.64402 q^{31} -1.00000 q^{32} -4.77791 q^{33} -3.00000 q^{34} +6.64402 q^{36} +4.53854 q^{37} -1.10548 q^{38} +7.85499 q^{41} +12.7495 q^{42} +4.00000 q^{43} -1.53854 q^{44} +6.74950 q^{46} +10.8550 q^{47} +3.10548 q^{48} +9.85499 q^{49} +9.31645 q^{51} +0.433057 q^{53} -11.3164 q^{54} +4.10548 q^{56} +3.43306 q^{57} +5.56694 q^{58} -13.7779 q^{59} -14.3164 q^{61} +8.64402 q^{62} -27.2769 q^{63} +1.00000 q^{64} +4.77791 q^{66} -3.74950 q^{67} +3.00000 q^{68} -20.9605 q^{69} -6.64402 q^{72} +10.7779 q^{73} -4.53854 q^{74} +1.10548 q^{76} +6.31645 q^{77} -6.53854 q^{79} +15.2110 q^{81} -7.85499 q^{82} -4.46146 q^{83} -12.7495 q^{84} -4.00000 q^{86} -17.2880 q^{87} +1.53854 q^{88} -1.85499 q^{89} -6.74950 q^{92} -26.8439 q^{93} -10.8550 q^{94} -3.10548 q^{96} -3.78903 q^{97} -9.85499 q^{98} -10.2221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{16} + 9 q^{17} - 9 q^{18} - 6 q^{19} - 18 q^{21} + 3 q^{22} + 6 q^{27} - 3 q^{28} - 9 q^{29} - 15 q^{31} - 3 q^{32} + 12 q^{33}+ \cdots - 57 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^9 - 3 * q^11 + 3 * q^14 + 3 * q^16 + 9 * q^17 - 9 * q^18 - 6 * q^19 - 18 * q^21 + 3 * q^22 + 6 * q^27 - 3 * q^28 - 9 * q^29 - 15 * q^31 - 3 * q^32 + 12 * q^33 - 9 * q^34 + 9 * q^36 + 12 * q^37 + 6 * q^38 - 6 * q^41 + 18 * q^42 + 12 * q^43 - 3 * q^44 + 3 * q^47 + 9 * q^53 - 6 * q^54 + 3 * q^56 + 18 * q^57 + 9 * q^58 - 15 * q^59 - 15 * q^61 + 15 * q^62 - 15 * q^63 + 3 * q^64 - 12 * q^66 + 9 * q^67 + 9 * q^68 - 24 * q^69 - 9 * q^72 + 6 * q^73 - 12 * q^74 - 6 * q^76 - 9 * q^77 - 18 * q^79 + 27 * q^81 + 6 * q^82 - 15 * q^83 - 18 * q^84 - 12 * q^86 - 30 * q^87 + 3 * q^88 + 24 * q^89 - 6 * q^93 - 3 * q^94 - 30 * q^97 - 57 * q^99

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