LMFDB - Embedded newform 8450.2.a.cx.1.7

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:	$1$
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			1.7
Root			$-2.90009$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.303663 q^{3} +1.00000 q^{4} +0.303663 q^{6} -1.08593 q^{7} +1.00000 q^{8} -2.90779 q^{9} -0.403887 q^{11} +0.303663 q^{12} -1.08593 q^{14} +1.00000 q^{16} +5.21387 q^{17} -2.90779 q^{18} +0.844510 q^{19} -0.329755 q^{21} -0.403887 q^{22} +1.76217 q^{23} +0.303663 q^{24} -1.79397 q^{27} -1.08593 q^{28} -5.96747 q^{29} -5.25338 q^{31} +1.00000 q^{32} -0.122645 q^{33} +5.21387 q^{34} -2.90779 q^{36} -4.42321 q^{37} +0.844510 q^{38} +3.20196 q^{41} -0.329755 q^{42} -11.4865 q^{43} -0.403887 q^{44} +1.76217 q^{46} +11.5048 q^{47} +0.303663 q^{48} -5.82076 q^{49} +1.58326 q^{51} -10.7632 q^{53} -1.79397 q^{54} -1.08593 q^{56} +0.256446 q^{57} -5.96747 q^{58} -7.28501 q^{59} -2.64135 q^{61} -5.25338 q^{62} +3.15764 q^{63} +1.00000 q^{64} -0.122645 q^{66} +10.5992 q^{67} +5.21387 q^{68} +0.535105 q^{69} -12.4325 q^{71} -2.90779 q^{72} -2.92837 q^{73} -4.42321 q^{74} +0.844510 q^{76} +0.438592 q^{77} +15.8880 q^{79} +8.17860 q^{81} +3.20196 q^{82} -12.6626 q^{83} -0.329755 q^{84} -11.4865 q^{86} -1.81210 q^{87} -0.403887 q^{88} +6.99770 q^{89} +1.76217 q^{92} -1.59525 q^{93} +11.5048 q^{94} +0.303663 q^{96} +9.47384 q^{97} -5.82076 q^{98} +1.17442 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.303663	0.175320	0.0876598	−	0.996150i	$-0.472061\pi$
$3$	0.303663	0.175320	0.0876598	+	0.996150i	$0.472061\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.303663	0.123970
$7$	−1.08593	−0.410441	−0.205221	−	0.978716i	$-0.565791\pi$
$7$	−1.08593	−0.410441	−0.205221	+	0.978716i	$0.565791\pi$
$8$	1.00000	0.353553
$9$	−2.90779	−0.969263
$10$	0	0
$11$	−0.403887	−0.121777	−0.0608883	−	0.998145i	$-0.519393\pi$
$11$	−0.403887	−0.121777	−0.0608883	+	0.998145i	$0.519393\pi$
$12$	0.303663	0.0876598
$13$	0	0
$13$	0	0
$14$	−1.08593	−0.290226
$15$	0	0
$16$	1.00000	0.250000
$17$	5.21387	1.26455	0.632274	−	0.774745i	$-0.282122\pi$
$17$	5.21387	1.26455	0.632274	+	0.774745i	$0.282122\pi$
$18$	−2.90779	−0.685372
$19$	0.844510	0.193744	0.0968719	−	0.995297i	$-0.469116\pi$
$19$	0.844510	0.193744	0.0968719	+	0.995297i	$0.469116\pi$
$20$	0	0
$21$	−0.329755	−0.0719585
$22$	−0.403887	−0.0861091
$23$	1.76217	0.367438	0.183719	−	0.982979i	$-0.441186\pi$
$23$	1.76217	0.367438	0.183719	+	0.982979i	$0.441186\pi$
$24$	0.303663	0.0619849
$25$	0	0
$26$	0	0
$27$	−1.79397	−0.345251
$28$	−1.08593	−0.205221
$29$	−5.96747	−1.10813	−0.554066	−	0.832473i	$-0.686924\pi$
$29$	−5.96747	−1.10813	−0.554066	+	0.832473i	$0.686924\pi$
$30$	0	0
$31$	−5.25338	−0.943534	−0.471767	−	0.881723i	$-0.656384\pi$
$31$	−5.25338	−0.943534	−0.471767	+	0.881723i	$0.656384\pi$
$32$	1.00000	0.176777
$33$	−0.122645	−0.0213498
$34$	5.21387	0.894171
$35$	0	0
$36$	−2.90779	−0.484632
$37$	−4.42321	−0.727171	−0.363586	−	0.931561i	$-0.618448\pi$
$37$	−4.42321	−0.727171	−0.363586	+	0.931561i	$0.618448\pi$
$38$	0.844510	0.136998
$39$	0	0
$40$	0	0
$41$	3.20196	0.500062	0.250031	−	0.968238i	$-0.419559\pi$
$41$	3.20196	0.500062	0.250031	+	0.968238i	$0.419559\pi$
$42$	−0.329755	−0.0508823
$43$	−11.4865	−1.75168	−0.875840	−	0.482601i	$-0.839692\pi$
$43$	−11.4865	−1.75168	−0.875840	+	0.482601i	$0.839692\pi$
$44$	−0.403887	−0.0608883
$45$	0	0
$46$	1.76217	0.259818
$47$	11.5048	1.67814	0.839071	−	0.544022i	$-0.183099\pi$
$47$	11.5048	1.67814	0.839071	+	0.544022i	$0.183099\pi$
$48$	0.303663	0.0438299
$49$	−5.82076	−0.831538
$50$	0	0
$51$	1.58326	0.221700
$52$	0	0
$53$	−10.7632	−1.47844	−0.739221	−	0.673463i	$-0.764806\pi$
$53$	−10.7632	−1.47844	−0.739221	+	0.673463i	$0.764806\pi$
$54$	−1.79397	−0.244129
$55$	0	0
$56$	−1.08593	−0.145113
$57$	0.256446	0.0339671
$58$	−5.96747	−0.783567
$59$	−7.28501	−0.948428	−0.474214	−	0.880410i	$-0.657268\pi$
$59$	−7.28501	−0.948428	−0.474214	+	0.880410i	$0.657268\pi$
$60$	0	0
$61$	−2.64135	−0.338190	−0.169095	−	0.985600i	$-0.554084\pi$
$61$	−2.64135	−0.338190	−0.169095	+	0.985600i	$0.554084\pi$
$62$	−5.25338	−0.667179
$63$	3.15764	0.397826
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.122645	−0.0150966
$67$	10.5992	1.29490	0.647452	−	0.762106i	$-0.275835\pi$
$67$	10.5992	1.29490	0.647452	+	0.762106i	$0.275835\pi$
$68$	5.21387	0.632274
$69$	0.535105	0.0644191
$70$	0	0
$71$	−12.4325	−1.47546	−0.737732	−	0.675094i	$-0.764103\pi$
$71$	−12.4325	−1.47546	−0.737732	+	0.675094i	$0.764103\pi$
$72$	−2.90779	−0.342686
$73$	−2.92837	−0.342740	−0.171370	−	0.985207i	$-0.554819\pi$
$73$	−2.92837	−0.342740	−0.171370	+	0.985207i	$0.554819\pi$
$74$	−4.42321	−0.514188
$75$	0	0
$76$	0.844510	0.0968719
$77$	0.438592	0.0499822
$78$	0	0
$79$	15.8880	1.78754	0.893772	−	0.448522i	$-0.148049\pi$
$79$	15.8880	1.78754	0.893772	+	0.448522i	$0.148049\pi$
$80$	0	0
$81$	8.17860	0.908734
$82$	3.20196	0.353597
$83$	−12.6626	−1.38990	−0.694952	−	0.719056i	$-0.744574\pi$
$83$	−12.6626	−1.38990	−0.694952	+	0.719056i	$0.744574\pi$
$84$	−0.329755	−0.0359792
$85$	0	0
$86$	−11.4865	−1.23863
$87$	−1.81210	−0.194277
$88$	−0.403887	−0.0430545
$89$	6.99770	0.741755	0.370877	−	0.928682i	$-0.379057\pi$
$89$	6.99770	0.741755	0.370877	+	0.928682i	$0.379057\pi$
$90$	0	0
$91$	0	0
$92$	1.76217	0.183719
$93$	−1.59525	−0.165420
$94$	11.5048	1.18663
$95$	0	0
$96$	0.303663	0.0309924
$97$	9.47384	0.961923	0.480961	−	0.876742i	$-0.340288\pi$
$97$	9.47384	0.961923	0.480961	+	0.876742i	$0.340288\pi$
$98$	−5.82076	−0.587986
$99$	1.17442	0.118034
$100$	0	0
$101$	6.61025	0.657744	0.328872	−	0.944374i	$-0.393331\pi$
$101$	6.61025	0.657744	0.328872	+	0.944374i	$0.393331\pi$
$102$	1.58326	0.156766
$103$	−9.21082	−0.907569	−0.453784	−	0.891111i	$-0.649926\pi$
$103$	−9.21082	−0.907569	−0.453784	+	0.891111i	$0.649926\pi$
$104$	0	0
$105$	0	0
$106$	−10.7632	−1.04542
$107$	−6.97470	−0.674270	−0.337135	−	0.941456i	$-0.609458\pi$
$107$	−6.97470	−0.674270	−0.337135	+	0.941456i	$0.609458\pi$
$108$	−1.79397	−0.172625
$109$	3.29363	0.315473	0.157736	−	0.987481i	$-0.449580\pi$
$109$	3.29363	0.315473	0.157736	+	0.987481i	$0.449580\pi$
$110$	0	0
$111$	−1.34316	−0.127487
$112$	−1.08593	−0.102610
$113$	10.0934	0.949508	0.474754	−	0.880119i	$-0.342537\pi$
$113$	10.0934	0.949508	0.474754	+	0.880119i	$0.342537\pi$
$114$	0.256446	0.0240184
$115$	0	0
$116$	−5.96747	−0.554066
$117$	0	0
$118$	−7.28501	−0.670640
$119$	−5.66187	−0.519023
$120$	0	0
$121$	−10.8369	−0.985170
$122$	−2.64135	−0.239137
$123$	0.972316	0.0876708
$124$	−5.25338	−0.471767
$125$	0	0
$126$	3.15764	0.281305
$127$	−6.06998	−0.538624	−0.269312	−	0.963053i	$-0.586796\pi$
$127$	−6.06998	−0.538624	−0.269312	+	0.963053i	$0.586796\pi$
$128$	1.00000	0.0883883
$129$	−3.48803	−0.307104
$130$	0	0
$131$	0.534787	0.0467246	0.0233623	−	0.999727i	$-0.492563\pi$
$131$	0.534787	0.0467246	0.0233623	+	0.999727i	$0.492563\pi$
$132$	−0.122645	−0.0106749
$133$	−0.917075	−0.0795205
$134$	10.5992	0.915636
$135$	0	0
$136$	5.21387	0.447085
$137$	−11.9809	−1.02360	−0.511799	−	0.859105i	$-0.671021\pi$
$137$	−11.9809	−1.02360	−0.511799	+	0.859105i	$0.671021\pi$
$138$	0.535105	0.0455512
$139$	2.69002	0.228165	0.114082	−	0.993471i	$-0.463607\pi$
$139$	2.69002	0.228165	0.114082	+	0.993471i	$0.463607\pi$
$140$	0	0
$141$	3.49357	0.294211
$142$	−12.4325	−1.04331
$143$	0	0
$144$	−2.90779	−0.242316
$145$	0	0
$146$	−2.92837	−0.242354
$147$	−1.76755	−0.145785
$148$	−4.42321	−0.363586
$149$	−17.5424	−1.43713	−0.718565	−	0.695460i	$-0.755201\pi$
$149$	−17.5424	−1.43713	−0.718565	+	0.695460i	$0.755201\pi$
$150$	0	0
$151$	17.5381	1.42723	0.713615	−	0.700538i	$-0.247057\pi$
$151$	17.5381	1.42723	0.713615	+	0.700538i	$0.247057\pi$
$152$	0.844510	0.0684988
$153$	−15.1608	−1.22568
$154$	0.438592	0.0353427
$155$	0	0
$156$	0	0
$157$	−10.5663	−0.843279	−0.421639	−	0.906764i	$-0.638545\pi$
$157$	−10.5663	−0.843279	−0.421639	+	0.906764i	$0.638545\pi$
$158$	15.8880	1.26398
$159$	−3.26839	−0.259200
$160$	0	0
$161$	−1.91359	−0.150812
$162$	8.17860	0.642572
$163$	−19.0637	−1.49318	−0.746592	−	0.665282i	$-0.768311\pi$
$163$	−19.0637	−1.49318	−0.746592	+	0.665282i	$0.768311\pi$
$164$	3.20196	0.250031
$165$	0	0
$166$	−12.6626	−0.982811
$167$	10.6825	0.826637	0.413319	−	0.910587i	$-0.364370\pi$
$167$	10.6825	0.826637	0.413319	+	0.910587i	$0.364370\pi$
$168$	−0.329755	−0.0254412
$169$	0	0
$170$	0	0
$171$	−2.45566	−0.187789
$172$	−11.4865	−0.875840
$173$	−15.8983	−1.20872	−0.604362	−	0.796710i	$-0.706572\pi$
$173$	−15.8983	−1.20872	−0.604362	+	0.796710i	$0.706572\pi$
$174$	−1.81210	−0.137375
$175$	0	0
$176$	−0.403887	−0.0304442
$177$	−2.21219	−0.166278
$178$	6.99770	0.524500
$179$	−25.2779	−1.88936	−0.944680	−	0.327995i	$-0.893627\pi$
$179$	−25.2779	−1.88936	−0.944680	+	0.327995i	$0.893627\pi$
$180$	0	0
$181$	17.8082	1.32367	0.661837	−	0.749648i	$-0.269777\pi$
$181$	17.8082	1.32367	0.661837	+	0.749648i	$0.269777\pi$
$182$	0	0
$183$	−0.802079	−0.0592914
$184$	1.76217	0.129909
$185$	0	0
$186$	−1.59525	−0.116970
$187$	−2.10581	−0.153992
$188$	11.5048	0.839071
$189$	1.94812	0.141705
$190$	0	0
$191$	−10.4612	−0.756947	−0.378474	−	0.925612i	$-0.623551\pi$
$191$	−10.4612	−0.756947	−0.378474	+	0.925612i	$0.623551\pi$
$192$	0.303663	0.0219150
$193$	14.6561	1.05497	0.527483	−	0.849565i	$-0.323136\pi$
$193$	14.6561	1.05497	0.527483	+	0.849565i	$0.323136\pi$
$194$	9.47384	0.680182
$195$	0	0
$196$	−5.82076	−0.415769
$197$	−10.3360	−0.736406	−0.368203	−	0.929745i	$-0.620027\pi$
$197$	−10.3360	−0.736406	−0.368203	+	0.929745i	$0.620027\pi$
$198$	1.17442	0.0834623
$199$	5.21516	0.369693	0.184846	−	0.982767i	$-0.440821\pi$
$199$	5.21516	0.369693	0.184846	+	0.982767i	$0.440821\pi$
$200$	0	0
$201$	3.21860	0.227022
$202$	6.61025	0.465096
$203$	6.48023	0.454823
$204$	1.58326	0.110850
$205$	0	0
$206$	−9.21082	−0.641748
$207$	−5.12402	−0.356144
$208$	0	0
$209$	−0.341087	−0.0235935
$210$	0	0
$211$	−12.3086	−0.847361	−0.423681	−	0.905812i	$-0.639262\pi$
$211$	−12.3086	−0.847361	−0.423681	+	0.905812i	$0.639262\pi$
$212$	−10.7632	−0.739221
$213$	−3.77528	−0.258678
$214$	−6.97470	−0.476781
$215$	0	0
$216$	−1.79397	−0.122065
$217$	5.70478	0.387266
$218$	3.29363	0.223073
$219$	−0.889236	−0.0600890
$220$	0	0
$221$	0	0
$222$	−1.34316	−0.0901472
$223$	9.35143	0.626218	0.313109	−	0.949717i	$-0.398629\pi$
$223$	9.35143	0.626218	0.313109	+	0.949717i	$0.398629\pi$
$224$	−1.08593	−0.0725565
$225$	0	0
$226$	10.0934	0.671403
$227$	−13.1715	−0.874224	−0.437112	−	0.899407i	$-0.643999\pi$
$227$	−13.1715	−0.874224	−0.437112	+	0.899407i	$0.643999\pi$
$228$	0.256446	0.0169836
$229$	−13.6752	−0.903680	−0.451840	−	0.892099i	$-0.649232\pi$
$229$	−13.6752	−0.903680	−0.451840	+	0.892099i	$0.649232\pi$
$230$	0	0
$231$	0.133184	0.00876286
$232$	−5.96747	−0.391784
$233$	−27.7357	−1.81703	−0.908514	−	0.417855i	$-0.862782\pi$
$233$	−27.7357	−1.81703	−0.908514	+	0.417855i	$0.862782\pi$
$234$	0	0
$235$	0	0
$236$	−7.28501	−0.474214
$237$	4.82460	0.313392
$238$	−5.66187	−0.367005
$239$	−17.7552	−1.14849	−0.574243	−	0.818685i	$-0.694703\pi$
$239$	−17.7552	−1.14849	−0.574243	+	0.818685i	$0.694703\pi$
$240$	0	0
$241$	−3.83937	−0.247316	−0.123658	−	0.992325i	$-0.539463\pi$
$241$	−3.83937	−0.247316	−0.123658	+	0.992325i	$0.539463\pi$
$242$	−10.8369	−0.696621
$243$	7.86546	0.504569
$244$	−2.64135	−0.169095
$245$	0	0
$246$	0.972316	0.0619926
$247$	0	0
$248$	−5.25338	−0.333590
$249$	−3.84517	−0.243678
$250$	0	0
$251$	14.4338	0.911054	0.455527	−	0.890222i	$-0.349451\pi$
$251$	14.4338	0.911054	0.455527	+	0.890222i	$0.349451\pi$
$252$	3.15764	0.198913
$253$	−0.711719	−0.0447454
$254$	−6.06998	−0.380865
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.0755	−0.690868	−0.345434	−	0.938443i	$-0.612268\pi$
$257$	−11.0755	−0.690868	−0.345434	+	0.938443i	$0.612268\pi$
$258$	−3.48803	−0.217155
$259$	4.80328	0.298461
$260$	0	0
$261$	17.3521	1.07407
$262$	0.534787	0.0330393
$263$	−1.74154	−0.107388	−0.0536941	−	0.998557i	$-0.517100\pi$
$263$	−1.74154	−0.107388	−0.0536941	+	0.998557i	$0.517100\pi$
$264$	−0.122645	−0.00754831
$265$	0	0
$266$	−0.917075	−0.0562295
$267$	2.12494	0.130044
$268$	10.5992	0.647452
$269$	−10.0411	−0.612220	−0.306110	−	0.951996i	$-0.599027\pi$
$269$	−10.0411	−0.612220	−0.306110	+	0.951996i	$0.599027\pi$
$270$	0	0
$271$	10.3097	0.626271	0.313136	−	0.949708i	$-0.398621\pi$
$271$	10.3097	0.626271	0.313136	+	0.949708i	$0.398621\pi$
$272$	5.21387	0.316137
$273$	0	0
$274$	−11.9809	−0.723794
$275$	0	0
$276$	0.535105	0.0322096
$277$	−29.7990	−1.79045	−0.895224	−	0.445616i	$-0.852985\pi$
$277$	−29.7990	−1.79045	−0.895224	+	0.445616i	$0.852985\pi$
$278$	2.69002	0.161337
$279$	15.2757	0.914533
$280$	0	0
$281$	17.5437	1.04657	0.523285	−	0.852158i	$-0.324706\pi$
$281$	17.5437	1.04657	0.523285	+	0.852158i	$0.324706\pi$
$282$	3.49357	0.208039
$283$	−25.9299	−1.54137	−0.770687	−	0.637214i	$-0.780087\pi$
$283$	−25.9299	−1.54137	−0.770687	+	0.637214i	$0.780087\pi$
$284$	−12.4325	−0.737732
$285$	0	0
$286$	0	0
$287$	−3.47709	−0.205246
$288$	−2.90779	−0.171343
$289$	10.1844	0.599083
$290$	0	0
$291$	2.87685	0.168644
$292$	−2.92837	−0.171370
$293$	−26.1627	−1.52844	−0.764221	−	0.644954i	$-0.776876\pi$
$293$	−26.1627	−1.52844	−0.764221	+	0.644954i	$0.776876\pi$
$294$	−1.76755	−0.103086
$295$	0	0
$296$	−4.42321	−0.257094
$297$	0.724564	0.0420434
$298$	−17.5424	−1.01620
$299$	0	0
$300$	0	0
$301$	12.4735	0.718962
$302$	17.5381	1.00920
$303$	2.00729	0.115316
$304$	0.844510	0.0484360
$305$	0	0
$306$	−15.1608	−0.866687
$307$	−2.68863	−0.153448	−0.0767240	−	0.997052i	$-0.524446\pi$
$307$	−2.68863	−0.153448	−0.0767240	+	0.997052i	$0.524446\pi$
$308$	0.438592	0.0249911
$309$	−2.79698	−0.159115
$310$	0	0
$311$	−31.0278	−1.75943	−0.879713	−	0.475506i	$-0.842265\pi$
$311$	−31.0278	−1.75943	−0.879713	+	0.475506i	$0.842265\pi$
$312$	0	0
$313$	−14.7399	−0.833151	−0.416575	−	0.909101i	$-0.636770\pi$
$313$	−14.7399	−0.833151	−0.416575	+	0.909101i	$0.636770\pi$
$314$	−10.5663	−0.596288
$315$	0	0
$316$	15.8880	0.893772
$317$	−14.7416	−0.827971	−0.413985	−	0.910284i	$-0.635864\pi$
$317$	−14.7416	−0.827971	−0.413985	+	0.910284i	$0.635864\pi$
$318$	−3.26839	−0.183282
$319$	2.41019	0.134944
$320$	0	0
$321$	−2.11796	−0.118213
$322$	−1.91359	−0.106640
$323$	4.40316	0.244999
$324$	8.17860	0.454367
$325$	0	0
$326$	−19.0637	−1.05584
$327$	1.00015	0.0553086
$328$	3.20196	0.176799
$329$	−12.4933	−0.688779
$330$	0	0
$331$	−9.55973	−0.525450	−0.262725	−	0.964871i	$-0.584621\pi$
$331$	−9.55973	−0.525450	−0.262725	+	0.964871i	$0.584621\pi$
$332$	−12.6626	−0.694952
$333$	12.8618	0.704820
$334$	10.6825	0.584521
$335$	0	0
$336$	−0.329755	−0.0179896
$337$	−7.58566	−0.413217	−0.206609	−	0.978424i	$-0.566243\pi$
$337$	−7.58566	−0.413217	−0.206609	+	0.978424i	$0.566243\pi$
$338$	0	0
$339$	3.06499	0.166467
$340$	0	0
$341$	2.12177	0.114900
$342$	−2.45566	−0.132787
$343$	13.9224	0.751739
$344$	−11.4865	−0.619313
$345$	0	0
$346$	−15.8983	−0.854697
$347$	8.88467	0.476954	0.238477	−	0.971148i	$-0.423352\pi$
$347$	8.88467	0.476954	0.238477	+	0.971148i	$0.423352\pi$
$348$	−1.81210	−0.0971386
$349$	22.3237	1.19496	0.597480	−	0.801884i	$-0.296169\pi$
$349$	22.3237	1.19496	0.597480	+	0.801884i	$0.296169\pi$
$350$	0	0
$351$	0	0
$352$	−0.403887	−0.0215273
$353$	31.0186	1.65095	0.825476	−	0.564438i	$-0.190907\pi$
$353$	31.0186	1.65095	0.825476	+	0.564438i	$0.190907\pi$
$354$	−2.21219	−0.117576
$355$	0	0
$356$	6.99770	0.370877
$357$	−1.71930	−0.0909950
$358$	−25.2779	−1.33598
$359$	15.1402	0.799072	0.399536	−	0.916718i	$-0.369171\pi$
$359$	15.1402	0.799072	0.399536	+	0.916718i	$0.369171\pi$
$360$	0	0
$361$	−18.2868	−0.962463
$362$	17.8082	0.935979
$363$	−3.29075	−0.172720
$364$	0	0
$365$	0	0
$366$	−0.802079	−0.0419253
$367$	20.8201	1.08680	0.543401	−	0.839474i	$-0.317137\pi$
$367$	20.8201	1.08680	0.543401	+	0.839474i	$0.317137\pi$
$368$	1.76217	0.0918595
$369$	−9.31063	−0.484692
$370$	0	0
$371$	11.6881	0.606814
$372$	−1.59525	−0.0827100
$373$	−0.901633	−0.0466848	−0.0233424	−	0.999728i	$-0.507431\pi$
$373$	−0.901633	−0.0466848	−0.0233424	+	0.999728i	$0.507431\pi$
$374$	−2.10581	−0.108889
$375$	0	0
$376$	11.5048	0.593313
$377$	0	0
$378$	1.94812	0.100201
$379$	−15.7125	−0.807096	−0.403548	−	0.914959i	$-0.632223\pi$
$379$	−15.7125	−0.807096	−0.403548	+	0.914959i	$0.632223\pi$
$380$	0	0
$381$	−1.84323	−0.0944314
$382$	−10.4612	−0.535242
$383$	38.0352	1.94351	0.971755	−	0.235993i	$-0.0758344\pi$
$383$	38.0352	1.94351	0.971755	+	0.235993i	$0.0758344\pi$
$384$	0.303663	0.0154962
$385$	0	0
$386$	14.6561	0.745974
$387$	33.4004	1.69784
$388$	9.47384	0.480961
$389$	3.64514	0.184816	0.0924079	−	0.995721i	$-0.470544\pi$
$389$	3.64514	0.184816	0.0924079	+	0.995721i	$0.470544\pi$
$390$	0	0
$391$	9.18773	0.464643
$392$	−5.82076	−0.293993
$393$	0.162395	0.00819173
$394$	−10.3360	−0.520718
$395$	0	0
$396$	1.17442	0.0590168
$397$	1.08366	0.0543872	0.0271936	−	0.999630i	$-0.491343\pi$
$397$	1.08366	0.0543872	0.0271936	+	0.999630i	$0.491343\pi$
$398$	5.21516	0.261412
$399$	−0.278481	−0.0139415
$400$	0	0
$401$	6.16953	0.308091	0.154046	−	0.988064i	$-0.450770\pi$
$401$	6.16953	0.308091	0.154046	+	0.988064i	$0.450770\pi$
$402$	3.21860	0.160529
$403$	0	0
$404$	6.61025	0.328872
$405$	0	0
$406$	6.48023	0.321608
$407$	1.78648	0.0885524
$408$	1.58326	0.0783829
$409$	−19.8455	−0.981299	−0.490649	−	0.871357i	$-0.663240\pi$
$409$	−19.8455	−0.981299	−0.490649	+	0.871357i	$0.663240\pi$
$410$	0	0
$411$	−3.63816	−0.179457
$412$	−9.21082	−0.453784
$413$	7.91099	0.389274
$414$	−5.12402	−0.251832
$415$	0	0
$416$	0	0
$417$	0.816859	0.0400018
$418$	−0.341087	−0.0166831
$419$	4.45670	0.217724	0.108862	−	0.994057i	$-0.465279\pi$
$419$	4.45670	0.217724	0.108862	+	0.994057i	$0.465279\pi$
$420$	0	0
$421$	−16.7557	−0.816624	−0.408312	−	0.912842i	$-0.633882\pi$
$421$	−16.7557	−0.816624	−0.408312	+	0.912842i	$0.633882\pi$
$422$	−12.3086	−0.599175
$423$	−33.4534	−1.62656
$424$	−10.7632	−0.522708
$425$	0	0
$426$	−3.77528	−0.182913
$427$	2.86831	0.138807
$428$	−6.97470	−0.337135
$429$	0	0
$430$	0	0
$431$	17.3440	0.835433	0.417717	−	0.908577i	$-0.362830\pi$
$431$	17.3440	0.835433	0.417717	+	0.908577i	$0.362830\pi$
$432$	−1.79397	−0.0863126
$433$	4.41765	0.212299	0.106149	−	0.994350i	$-0.466148\pi$
$433$	4.41765	0.212299	0.106149	+	0.994350i	$0.466148\pi$
$434$	5.70478	0.273838
$435$	0	0
$436$	3.29363	0.157736
$437$	1.48817	0.0711889
$438$	−0.889236	−0.0424893
$439$	33.1534	1.58232	0.791162	−	0.611607i	$-0.209477\pi$
$439$	33.1534	1.58232	0.791162	+	0.611607i	$0.209477\pi$
$440$	0	0
$441$	16.9256	0.805979
$442$	0	0
$443$	−19.2632	−0.915224	−0.457612	−	0.889152i	$-0.651295\pi$
$443$	−19.2632	−0.915224	−0.457612	+	0.889152i	$0.651295\pi$
$444$	−1.34316	−0.0637437
$445$	0	0
$446$	9.35143	0.442803
$447$	−5.32697	−0.251957
$448$	−1.08593	−0.0513052
$449$	4.55734	0.215074	0.107537	−	0.994201i	$-0.465704\pi$
$449$	4.55734	0.215074	0.107537	+	0.994201i	$0.465704\pi$
$450$	0	0
$451$	−1.29323	−0.0608959
$452$	10.0934	0.474754
$453$	5.32566	0.250222
$454$	−13.1715	−0.618170
$455$	0	0
$456$	0.256446	0.0120092
$457$	15.9598	0.746570	0.373285	−	0.927717i	$-0.378231\pi$
$457$	15.9598	0.746570	0.373285	+	0.927717i	$0.378231\pi$
$458$	−13.6752	−0.638998
$459$	−9.35354	−0.436586
$460$	0	0
$461$	22.2437	1.03599	0.517996	−	0.855383i	$-0.326678\pi$
$461$	22.2437	1.03599	0.517996	+	0.855383i	$0.326678\pi$
$462$	0.133184	0.00619628
$463$	31.0149	1.44138	0.720692	−	0.693256i	$-0.243824\pi$
$463$	31.0149	1.44138	0.720692	+	0.693256i	$0.243824\pi$
$464$	−5.96747	−0.277033
$465$	0	0
$466$	−27.7357	−1.28483
$467$	−0.841337	−0.0389324	−0.0194662	−	0.999811i	$-0.506197\pi$
$467$	−0.841337	−0.0389324	−0.0194662	+	0.999811i	$0.506197\pi$
$468$	0	0
$469$	−11.5100	−0.531482
$470$	0	0
$471$	−3.20858	−0.147843
$472$	−7.28501	−0.335320
$473$	4.63927	0.213314
$474$	4.82460	0.221601
$475$	0	0
$476$	−5.66187	−0.259512
$477$	31.2972	1.43300
$478$	−17.7552	−0.812102
$479$	7.63429	0.348819	0.174410	−	0.984673i	$-0.444198\pi$
$479$	7.63429	0.348819	0.174410	+	0.984673i	$0.444198\pi$
$480$	0	0
$481$	0	0
$482$	−3.83937	−0.174879
$483$	−0.581085	−0.0264403
$484$	−10.8369	−0.492585
$485$	0	0
$486$	7.86546	0.356784
$487$	−23.2958	−1.05563	−0.527817	−	0.849358i	$-0.676989\pi$
$487$	−23.2958	−1.05563	−0.527817	+	0.849358i	$0.676989\pi$
$488$	−2.64135	−0.119568
$489$	−5.78893	−0.261784
$490$	0	0
$491$	36.2972	1.63807	0.819034	−	0.573745i	$-0.194510\pi$
$491$	36.2972	1.63807	0.819034	+	0.573745i	$0.194510\pi$
$492$	0.972316	0.0438354
$493$	−31.1136	−1.40129
$494$	0	0
$495$	0	0
$496$	−5.25338	−0.235884
$497$	13.5008	0.605591
$498$	−3.84517	−0.172306
$499$	18.9398	0.847863	0.423932	−	0.905694i	$-0.360650\pi$
$499$	18.9398	0.847863	0.423932	+	0.905694i	$0.360650\pi$
$500$	0	0
$501$	3.24388	0.144926
$502$	14.4338	0.644213
$503$	35.7640	1.59464	0.797318	−	0.603560i	$-0.206251\pi$
$503$	35.7640	1.59464	0.797318	+	0.603560i	$0.206251\pi$
$504$	3.15764	0.140653
$505$	0	0
$506$	−0.711719	−0.0316398
$507$	0	0
$508$	−6.06998	−0.269312
$509$	3.11873	0.138235	0.0691176	−	0.997609i	$-0.477982\pi$
$509$	3.11873	0.138235	0.0691176	+	0.997609i	$0.477982\pi$
$510$	0	0
$511$	3.17999	0.140675
$512$	1.00000	0.0441942
$513$	−1.51503	−0.0668902
$514$	−11.0755	−0.488517
$515$	0	0
$516$	−3.48803	−0.153552
$517$	−4.64663	−0.204359
$518$	4.80328	0.211044
$519$	−4.82772	−0.211913
$520$	0	0
$521$	38.4309	1.68369	0.841844	−	0.539721i	$-0.181470\pi$
$521$	38.4309	1.68369	0.841844	+	0.539721i	$0.181470\pi$
$522$	17.3521	0.759483
$523$	23.8519	1.04297	0.521485	−	0.853260i	$-0.325378\pi$
$523$	23.8519	1.04297	0.521485	+	0.853260i	$0.325378\pi$
$524$	0.534787	0.0233623
$525$	0	0
$526$	−1.74154	−0.0759349
$527$	−27.3904	−1.19314
$528$	−0.122645	−0.00533746
$529$	−19.8948	−0.864989
$530$	0	0
$531$	21.1833	0.919276
$532$	−0.917075	−0.0397603
$533$	0	0
$534$	2.12494	0.0919551
$535$	0	0
$536$	10.5992	0.457818
$537$	−7.67595	−0.331242
$538$	−10.0411	−0.432905
$539$	2.35093	0.101262
$540$	0	0
$541$	5.36874	0.230820	0.115410	−	0.993318i	$-0.463182\pi$
$541$	5.36874	0.230820	0.115410	+	0.993318i	$0.463182\pi$
$542$	10.3097	0.442841
$543$	5.40769	0.232066
$544$	5.21387	0.223543
$545$	0	0
$546$	0	0
$547$	−28.1689	−1.20442	−0.602208	−	0.798339i	$-0.705712\pi$
$547$	−28.1689	−1.20442	−0.602208	+	0.798339i	$0.705712\pi$
$548$	−11.9809	−0.511799
$549$	7.68049	0.327795
$550$	0	0
$551$	−5.03959	−0.214694
$552$	0.535105	0.0227756
$553$	−17.2532	−0.733682
$554$	−29.7990	−1.26604
$555$	0	0
$556$	2.69002	0.114082
$557$	15.1505	0.641947	0.320974	−	0.947088i	$-0.395990\pi$
$557$	15.1505	0.641947	0.320974	+	0.947088i	$0.395990\pi$
$558$	15.2757	0.646672
$559$	0	0
$560$	0	0
$561$	−0.639457	−0.0269979
$562$	17.5437	0.740037
$563$	−32.5894	−1.37348	−0.686739	−	0.726904i	$-0.740958\pi$
$563$	−32.5894	−1.37348	−0.686739	+	0.726904i	$0.740958\pi$
$564$	3.49357	0.147106
$565$	0	0
$566$	−25.9299	−1.08992
$567$	−8.88136	−0.372982
$568$	−12.4325	−0.521655
$569$	21.7554	0.912033	0.456016	−	0.889971i	$-0.349276\pi$
$569$	21.7554	0.912033	0.456016	+	0.889971i	$0.349276\pi$
$570$	0	0
$571$	−24.2853	−1.01631	−0.508155	−	0.861266i	$-0.669672\pi$
$571$	−24.2853	−1.01631	−0.508155	+	0.861266i	$0.669672\pi$
$572$	0	0
$573$	−3.17668	−0.132708
$574$	−3.47709	−0.145131
$575$	0	0
$576$	−2.90779	−0.121158
$577$	23.1332	0.963048	0.481524	−	0.876433i	$-0.340083\pi$
$577$	23.1332	0.963048	0.481524	+	0.876433i	$0.340083\pi$
$578$	10.1844	0.423616
$579$	4.45050	0.184956
$580$	0	0
$581$	13.7507	0.570474
$582$	2.87685	0.119249
$583$	4.34713	0.180040
$584$	−2.92837	−0.121177
$585$	0	0
$586$	−26.1627	−1.08077
$587$	−4.60818	−0.190200	−0.0950999	−	0.995468i	$-0.530317\pi$
$587$	−4.60818	−0.190200	−0.0950999	+	0.995468i	$0.530317\pi$
$588$	−1.76755	−0.0728925
$589$	−4.43653	−0.182804
$590$	0	0
$591$	−3.13864	−0.129106
$592$	−4.42321	−0.181793
$593$	1.27629	0.0524111	0.0262056	−	0.999657i	$-0.491658\pi$
$593$	1.27629	0.0524111	0.0262056	+	0.999657i	$0.491658\pi$
$594$	0.724564	0.0297292
$595$	0	0
$596$	−17.5424	−0.718565
$597$	1.58365	0.0648144
$598$	0	0
$599$	21.3539	0.872497	0.436249	−	0.899826i	$-0.356307\pi$
$599$	21.3539	0.872497	0.436249	+	0.899826i	$0.356307\pi$
$600$	0	0
$601$	−2.74727	−0.112063	−0.0560317	−	0.998429i	$-0.517845\pi$
$601$	−2.74727	−0.112063	−0.0560317	+	0.998429i	$0.517845\pi$
$602$	12.4735	0.508383
$603$	−30.8204	−1.25510
$604$	17.5381	0.713615
$605$	0	0
$606$	2.00729	0.0815404
$607$	−22.9135	−0.930030	−0.465015	−	0.885303i	$-0.653951\pi$
$607$	−22.9135	−0.930030	−0.465015	+	0.885303i	$0.653951\pi$
$608$	0.844510	0.0342494
$609$	1.96780	0.0797394
$610$	0	0
$611$	0	0
$612$	−15.1608	−0.612840
$613$	16.5575	0.668752	0.334376	−	0.942440i	$-0.391474\pi$
$613$	16.5575	0.668752	0.334376	+	0.942440i	$0.391474\pi$
$614$	−2.68863	−0.108504
$615$	0	0
$616$	0.438592	0.0176714
$617$	11.3693	0.457712	0.228856	−	0.973460i	$-0.426502\pi$
$617$	11.3693	0.457712	0.228856	+	0.973460i	$0.426502\pi$
$618$	−2.79698	−0.112511
$619$	47.1109	1.89355	0.946773	−	0.321901i	$-0.104322\pi$
$619$	47.1109	1.89355	0.946773	+	0.321901i	$0.104322\pi$
$620$	0	0
$621$	−3.16129	−0.126858
$622$	−31.0278	−1.24410
$623$	−7.59898	−0.304447
$624$	0	0
$625$	0	0
$626$	−14.7399	−0.589127
$627$	−0.103575	−0.00413640
$628$	−10.5663	−0.421639
$629$	−23.0620	−0.919543
$630$	0	0
$631$	8.39389	0.334156	0.167078	−	0.985944i	$-0.446567\pi$
$631$	8.39389	0.334156	0.167078	+	0.985944i	$0.446567\pi$
$632$	15.8880	0.631992
$633$	−3.73767	−0.148559
$634$	−14.7416	−0.585464
$635$	0	0
$636$	−3.26839	−0.129600
$637$	0	0
$638$	2.41019	0.0954201
$639$	36.1510	1.43011
$640$	0	0
$641$	−36.5728	−1.44454	−0.722268	−	0.691613i	$-0.756901\pi$
$641$	−36.5728	−1.44454	−0.722268	+	0.691613i	$0.756901\pi$
$642$	−2.11796	−0.0835891
$643$	−46.0372	−1.81553	−0.907766	−	0.419478i	$-0.862213\pi$
$643$	−46.0372	−1.81553	−0.907766	+	0.419478i	$0.862213\pi$
$644$	−1.91359	−0.0754059
$645$	0	0
$646$	4.40316	0.173240
$647$	−27.8549	−1.09509	−0.547545	−	0.836776i	$-0.684437\pi$
$647$	−27.8549	−1.09509	−0.547545	+	0.836776i	$0.684437\pi$
$648$	8.17860	0.321286
$649$	2.94232	0.115496
$650$	0	0
$651$	1.73233	0.0678953
$652$	−19.0637	−0.746592
$653$	−4.98456	−0.195061	−0.0975304	−	0.995233i	$-0.531094\pi$
$653$	−4.98456	−0.195061	−0.0975304	+	0.995233i	$0.531094\pi$
$654$	1.00015	0.0391091
$655$	0	0
$656$	3.20196	0.125016
$657$	8.51508	0.332205
$658$	−12.4933	−0.487040
$659$	27.8625	1.08537	0.542684	−	0.839937i	$-0.317408\pi$
$659$	27.8625	1.08537	0.542684	+	0.839937i	$0.317408\pi$
$660$	0	0
$661$	37.9173	1.47481	0.737406	−	0.675450i	$-0.236051\pi$
$661$	37.9173	1.47481	0.737406	+	0.675450i	$0.236051\pi$
$662$	−9.55973	−0.371549
$663$	0	0
$664$	−12.6626	−0.491405
$665$	0	0
$666$	12.8618	0.498383
$667$	−10.5157	−0.407170
$668$	10.6825	0.413319
$669$	2.83968	0.109788
$670$	0	0
$671$	1.06681	0.0411837
$672$	−0.329755	−0.0127206
$673$	5.08337	0.195949	0.0979747	−	0.995189i	$-0.468764\pi$
$673$	5.08337	0.195949	0.0979747	+	0.995189i	$0.468764\pi$
$674$	−7.58566	−0.292189
$675$	0	0
$676$	0	0
$677$	40.3815	1.55199	0.775994	−	0.630740i	$-0.217249\pi$
$677$	40.3815	1.55199	0.775994	+	0.630740i	$0.217249\pi$
$678$	3.06499	0.117710
$679$	−10.2879	−0.394813
$680$	0	0
$681$	−3.99970	−0.153269
$682$	2.12177	0.0812468
$683$	14.8432	0.567959	0.283980	−	0.958830i	$-0.408345\pi$
$683$	14.8432	0.567959	0.283980	+	0.958830i	$0.408345\pi$
$684$	−2.45566	−0.0938944
$685$	0	0
$686$	13.9224	0.531560
$687$	−4.15264	−0.158433
$688$	−11.4865	−0.437920
$689$	0	0
$690$	0	0
$691$	31.8114	1.21016	0.605081	−	0.796164i	$-0.293141\pi$
$691$	31.8114	1.21016	0.605081	+	0.796164i	$0.293141\pi$
$692$	−15.8983	−0.604362
$693$	−1.27533	−0.0484459
$694$	8.88467	0.337257
$695$	0	0
$696$	−1.81210	−0.0686874
$697$	16.6946	0.632353
$698$	22.3237	0.844964
$699$	−8.42230	−0.318561
$700$	0	0
$701$	−2.75131	−0.103915	−0.0519577	−	0.998649i	$-0.516546\pi$
$701$	−2.75131	−0.103915	−0.0519577	+	0.998649i	$0.516546\pi$
$702$	0	0
$703$	−3.73544	−0.140885
$704$	−0.403887	−0.0152221
$705$	0	0
$706$	31.0186	1.16740
$707$	−7.17824	−0.269966
$708$	−2.21219	−0.0831390
$709$	28.7721	1.08056	0.540279	−	0.841486i	$-0.318319\pi$
$709$	28.7721	1.08056	0.540279	+	0.841486i	$0.318319\pi$
$710$	0	0
$711$	−46.1991	−1.73260
$712$	6.99770	0.262250
$713$	−9.25735	−0.346690
$714$	−1.71930	−0.0643432
$715$	0	0
$716$	−25.2779	−0.944680
$717$	−5.39158	−0.201352
$718$	15.1402	0.565029
$719$	33.1840	1.23756	0.618778	−	0.785566i	$-0.287628\pi$
$719$	33.1840	1.23756	0.618778	+	0.785566i	$0.287628\pi$
$720$	0	0
$721$	10.0023	0.372504
$722$	−18.2868	−0.680564
$723$	−1.16587	−0.0433593
$724$	17.8082	0.661837
$725$	0	0
$726$	−3.29075	−0.122131
$727$	−37.6129	−1.39499	−0.697493	−	0.716591i	$-0.745701\pi$
$727$	−37.6129	−1.39499	−0.697493	+	0.716591i	$0.745701\pi$
$728$	0	0
$729$	−22.1474	−0.820273
$730$	0	0
$731$	−59.8893	−2.21509
$732$	−0.802079	−0.0296457
$733$	−19.0778	−0.704654	−0.352327	−	0.935877i	$-0.614609\pi$
$733$	−19.0778	−0.704654	−0.352327	+	0.935877i	$0.614609\pi$
$734$	20.8201	0.768484
$735$	0	0
$736$	1.76217	0.0649545
$737$	−4.28090	−0.157689
$738$	−9.31063	−0.342729
$739$	−40.5137	−1.49032	−0.745160	−	0.666886i	$-0.767627\pi$
$739$	−40.5137	−1.49032	−0.745160	+	0.666886i	$0.767627\pi$
$740$	0	0
$741$	0	0
$742$	11.6881	0.429082
$743$	11.8075	0.433176	0.216588	−	0.976263i	$-0.430507\pi$
$743$	11.8075	0.433176	0.216588	+	0.976263i	$0.430507\pi$
$744$	−1.59525	−0.0584848
$745$	0	0
$746$	−0.901633	−0.0330111
$747$	36.8203	1.34718
$748$	−2.10581	−0.0769962
$749$	7.57401	0.276748
$750$	0	0
$751$	−14.0149	−0.511409	−0.255705	−	0.966755i	$-0.582307\pi$
$751$	−14.0149	−0.511409	−0.255705	+	0.966755i	$0.582307\pi$
$752$	11.5048	0.419536
$753$	4.38301	0.159726
$754$	0	0
$755$	0	0
$756$	1.94812	0.0708526
$757$	−31.2552	−1.13599	−0.567994	−	0.823032i	$-0.692280\pi$
$757$	−31.2552	−1.13599	−0.567994	+	0.823032i	$0.692280\pi$
$758$	−15.7125	−0.570703
$759$	−0.216122	−0.00784474
$760$	0	0
$761$	34.1575	1.23821	0.619103	−	0.785310i	$-0.287496\pi$
$761$	34.1575	1.23821	0.619103	+	0.785310i	$0.287496\pi$
$762$	−1.84323	−0.0667731
$763$	−3.57664	−0.129483
$764$	−10.4612	−0.378474
$765$	0	0
$766$	38.0352	1.37427
$767$	0	0
$768$	0.303663	0.0109575
$769$	44.2201	1.59462	0.797308	−	0.603572i	$-0.206256\pi$
$769$	44.2201	1.59462	0.797308	+	0.603572i	$0.206256\pi$
$770$	0	0
$771$	−3.36320	−0.121123
$772$	14.6561	0.527483
$773$	2.04145	0.0734259	0.0367129	−	0.999326i	$-0.488311\pi$
$773$	2.04145	0.0734259	0.0367129	+	0.999326i	$0.488311\pi$
$774$	33.4004	1.20055
$775$	0	0
$776$	9.47384	0.340091
$777$	1.45858	0.0523261
$778$	3.64514	0.130685
$779$	2.70409	0.0968840
$780$	0	0
$781$	5.02132	0.179677
$782$	9.18773	0.328552
$783$	10.7055	0.382583
$784$	−5.82076	−0.207884
$785$	0	0
$786$	0.162395	0.00579243
$787$	−3.39520	−0.121026	−0.0605130	−	0.998167i	$-0.519274\pi$
$787$	−3.39520	−0.121026	−0.0605130	+	0.998167i	$0.519274\pi$
$788$	−10.3360	−0.368203
$789$	−0.528841	−0.0188273
$790$	0	0
$791$	−10.9607	−0.389717
$792$	1.17442	0.0417312
$793$	0	0
$794$	1.08366	0.0384576
$795$	0	0
$796$	5.21516	0.184846
$797$	24.4659	0.866626	0.433313	−	0.901244i	$-0.357345\pi$
$797$	24.4659	0.866626	0.433313	+	0.901244i	$0.357345\pi$
$798$	−0.278481	−0.00985814
$799$	59.9843	2.12209
$800$	0	0
$801$	−20.3478	−0.718955
$802$	6.16953	0.217854
$803$	1.18273	0.0417377
$804$	3.21860	0.113511
$805$	0	0
$806$	0	0
$807$	−3.04912	−0.107334
$808$	6.61025	0.232548
$809$	−17.6948	−0.622117	−0.311058	−	0.950391i	$-0.600683\pi$
$809$	−17.6948	−0.622117	−0.311058	+	0.950391i	$0.600683\pi$
$810$	0	0
$811$	−2.82931	−0.0993505	−0.0496753	−	0.998765i	$-0.515819\pi$
$811$	−2.82931	−0.0993505	−0.0496753	+	0.998765i	$0.515819\pi$
$812$	6.48023	0.227411
$813$	3.13068	0.109798
$814$	1.78648	0.0626160
$815$	0	0
$816$	1.58326	0.0554251
$817$	−9.70050	−0.339377
$818$	−19.8455	−0.693883
$819$	0	0
$820$	0	0
$821$	−26.1300	−0.911942	−0.455971	−	0.889995i	$-0.650708\pi$
$821$	−26.1300	−0.911942	−0.455971	+	0.889995i	$0.650708\pi$
$822$	−3.63816	−0.126895
$823$	1.85316	0.0645970	0.0322985	−	0.999478i	$-0.489717\pi$
$823$	1.85316	0.0645970	0.0322985	+	0.999478i	$0.489717\pi$
$824$	−9.21082	−0.320874
$825$	0	0
$826$	7.91099	0.275258
$827$	−19.9886	−0.695072	−0.347536	−	0.937667i	$-0.612982\pi$
$827$	−19.9886	−0.695072	−0.347536	+	0.937667i	$0.612982\pi$
$828$	−5.12402	−0.178072
$829$	41.6506	1.44658	0.723292	−	0.690542i	$-0.242628\pi$
$829$	41.6506	1.44658	0.723292	+	0.690542i	$0.242628\pi$
$830$	0	0
$831$	−9.04884	−0.313901
$832$	0	0
$833$	−30.3487	−1.05152
$834$	0.816859	0.0282855
$835$	0	0
$836$	−0.341087	−0.0117967
$837$	9.42442	0.325756
$838$	4.45670	0.153954
$839$	−25.9421	−0.895622	−0.447811	−	0.894128i	$-0.647796\pi$
$839$	−25.9421	−0.895622	−0.447811	+	0.894128i	$0.647796\pi$
$840$	0	0
$841$	6.61068	0.227955
$842$	−16.7557	−0.577441
$843$	5.32737	0.183484
$844$	−12.3086	−0.423681
$845$	0	0
$846$	−33.4534	−1.15015
$847$	11.7680	0.404355
$848$	−10.7632	−0.369610
$849$	−7.87395	−0.270233
$850$	0	0
$851$	−7.79445	−0.267190
$852$	−3.77528	−0.129339
$853$	−33.6674	−1.15275	−0.576374	−	0.817186i	$-0.695533\pi$
$853$	−33.6674	−1.15275	−0.576374	+	0.817186i	$0.695533\pi$
$854$	2.86831	0.0981516
$855$	0	0
$856$	−6.97470	−0.238391
$857$	6.39008	0.218281	0.109140	−	0.994026i	$-0.465190\pi$
$857$	6.39008	0.218281	0.109140	+	0.994026i	$0.465190\pi$
$858$	0	0
$859$	29.6345	1.01112	0.505558	−	0.862792i	$-0.331287\pi$
$859$	29.6345	1.01112	0.505558	+	0.862792i	$0.331287\pi$
$860$	0	0
$861$	−1.05586	−0.0359837
$862$	17.3440	0.590741
$863$	−15.7244	−0.535264	−0.267632	−	0.963521i	$-0.586241\pi$
$863$	−15.7244	−0.535264	−0.267632	+	0.963521i	$0.586241\pi$
$864$	−1.79397	−0.0610323
$865$	0	0
$866$	4.41765	0.150118
$867$	3.09262	0.105031
$868$	5.70478	0.193633
$869$	−6.41698	−0.217681
$870$	0	0
$871$	0	0
$872$	3.29363	0.111537
$873$	−27.5479	−0.932356
$874$	1.48817	0.0503381
$875$	0	0
$876$	−0.889236	−0.0300445
$877$	30.3994	1.02651	0.513257	−	0.858235i	$-0.328439\pi$
$877$	30.3994	1.02651	0.513257	+	0.858235i	$0.328439\pi$
$878$	33.1534	1.11887
$879$	−7.94464	−0.267966
$880$	0	0
$881$	41.3933	1.39458	0.697288	−	0.716791i	$-0.254390\pi$
$881$	41.3933	1.39458	0.697288	+	0.716791i	$0.254390\pi$
$882$	16.9256	0.569913
$883$	24.4503	0.822819	0.411410	−	0.911451i	$-0.365037\pi$
$883$	24.4503	0.822819	0.411410	+	0.911451i	$0.365037\pi$
$884$	0	0
$885$	0	0
$886$	−19.2632	−0.647161
$887$	−37.1737	−1.24817	−0.624085	−	0.781357i	$-0.714528\pi$
$887$	−37.1737	−1.24817	−0.624085	+	0.781357i	$0.714528\pi$
$888$	−1.34316	−0.0450736
$889$	6.59155	0.221074
$890$	0	0
$891$	−3.30323	−0.110663
$892$	9.35143	0.313109
$893$	9.71589	0.325130
$894$	−5.32697	−0.178161
$895$	0	0
$896$	−1.08593	−0.0362782
$897$	0	0
$898$	4.55734	0.152081
$899$	31.3494	1.04556
$900$	0	0
$901$	−56.1180	−1.86956
$902$	−1.29323	−0.0430599
$903$	3.78775	0.126048
$904$	10.0934	0.335702
$905$	0	0
$906$	5.32566	0.176933
$907$	−32.4758	−1.07834	−0.539171	−	0.842196i	$-0.681262\pi$
$907$	−32.4758	−1.07834	−0.539171	+	0.842196i	$0.681262\pi$
$908$	−13.1715	−0.437112
$909$	−19.2212	−0.637527
$910$	0	0
$911$	16.8692	0.558900	0.279450	−	0.960160i	$-0.409848\pi$
$911$	16.8692	0.558900	0.279450	+	0.960160i	$0.409848\pi$
$912$	0.256446	0.00849178
$913$	5.11428	0.169258
$914$	15.9598	0.527905
$915$	0	0
$916$	−13.6752	−0.451840
$917$	−0.580739	−0.0191777
$918$	−9.35354	−0.308713
$919$	−5.75843	−0.189953	−0.0949765	−	0.995480i	$-0.530278\pi$
$919$	−5.75843	−0.189953	−0.0949765	+	0.995480i	$0.530278\pi$
$920$	0	0
$921$	−0.816435	−0.0269025
$922$	22.2437	0.732556
$923$	0	0
$924$	0.133184	0.00438143
$925$	0	0
$926$	31.0149	1.01921
$927$	26.7831	0.879673
$928$	−5.96747	−0.195892
$929$	−43.2786	−1.41992	−0.709962	−	0.704240i	$-0.751288\pi$
$929$	−43.2786	−1.41992	−0.709962	+	0.704240i	$0.751288\pi$
$930$	0	0
$931$	−4.91569	−0.161105
$932$	−27.7357	−0.908514
$933$	−9.42198	−0.308462
$934$	−0.841337	−0.0275294
$935$	0	0
$936$	0	0
$937$	13.9713	0.456424	0.228212	−	0.973611i	$-0.426712\pi$
$937$	13.9713	0.456424	0.228212	+	0.973611i	$0.426712\pi$
$938$	−11.5100	−0.375815
$939$	−4.47597	−0.146068
$940$	0	0
$941$	−27.2678	−0.888905	−0.444452	−	0.895803i	$-0.646602\pi$
$941$	−27.2678	−0.888905	−0.444452	+	0.895803i	$0.646602\pi$
$942$	−3.20858	−0.104541
$943$	5.64240	0.183742
$944$	−7.28501	−0.237107
$945$	0	0
$946$	4.63927	0.150836
$947$	11.4663	0.372605	0.186302	−	0.982492i	$-0.440350\pi$
$947$	11.4663	0.372605	0.186302	+	0.982492i	$0.440350\pi$
$948$	4.82460	0.156696
$949$	0	0
$950$	0	0
$951$	−4.47647	−0.145160
$952$	−5.66187	−0.183502
$953$	−19.2137	−0.622392	−0.311196	−	0.950346i	$-0.600730\pi$
$953$	−19.2137	−0.622392	−0.311196	+	0.950346i	$0.600730\pi$
$954$	31.2972	1.01328
$955$	0	0
$956$	−17.7552	−0.574243
$957$	0.731883	0.0236584
$958$	7.63429	0.246653
$959$	13.0104	0.420127
$960$	0	0
$961$	−3.40205	−0.109743
$962$	0	0
$963$	20.2810	0.653545
$964$	−3.83937	−0.123658
$965$	0	0
$966$	−0.581085	−0.0186961
$967$	40.2200	1.29339	0.646695	−	0.762749i	$-0.276151\pi$
$967$	40.2200	1.29339	0.646695	+	0.762749i	$0.276151\pi$
$968$	−10.8369	−0.348310
$969$	1.33708	0.0429531
$970$	0	0
$971$	40.8959	1.31241	0.656207	−	0.754581i	$-0.272160\pi$
$971$	40.8959	1.31241	0.656207	+	0.754581i	$0.272160\pi$
$972$	7.86546	0.252285
$973$	−2.92117	−0.0936483
$974$	−23.2958	−0.746446
$975$	0	0
$976$	−2.64135	−0.0845475
$977$	−19.5975	−0.626981	−0.313490	−	0.949591i	$-0.601498\pi$
$977$	−19.5975	−0.626981	−0.313490	+	0.949591i	$0.601498\pi$
$978$	−5.78893	−0.185110
$979$	−2.82628	−0.0903284
$980$	0	0
$981$	−9.57719	−0.305776
$982$	36.2972	1.15829
$983$	−11.7949	−0.376198	−0.188099	−	0.982150i	$-0.560233\pi$
$983$	−11.7949	−0.376198	−0.188099	+	0.982150i	$0.560233\pi$
$984$	0.972316	0.0309963
$985$	0	0
$986$	−31.1136	−0.990858
$987$	−3.79375	−0.120757
$988$	0	0
$989$	−20.2412	−0.643634
$990$	0	0
$991$	25.3114	0.804043	0.402022	−	0.915630i	$-0.368308\pi$
$991$	25.3114	0.804043	0.402022	+	0.915630i	$0.368308\pi$
$992$	−5.25338	−0.166795
$993$	−2.90293	−0.0921217
$994$	13.5008	0.428218
$995$	0	0
$996$	−3.84517	−0.121839
$997$	−18.7441	−0.593632	−0.296816	−	0.954935i	$-0.595925\pi$
$997$	−18.7441	−0.593632	−0.296816	+	0.954935i	$0.595925\pi$
$998$	18.9398	0.599530
$999$	7.93513	0.251056

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cx.1.7		9
5.2	odd	4		1690.2.b.f.339.12	yes	18
5.3	odd	4		1690.2.b.f.339.7	✓	18
5.4	even	2		8450.2.a.cw.1.3		9
13.12	even	2		8450.2.a.ct.1.7		9
65.8	even	4		1690.2.c.h.1689.11		18
65.12	odd	4		1690.2.b.g.339.3	yes	18
65.18	even	4		1690.2.c.g.1689.11		18
65.38	odd	4		1690.2.b.g.339.16	yes	18
65.47	even	4		1690.2.c.g.1689.8		18
65.57	even	4		1690.2.c.h.1689.8		18
65.64	even	2		8450.2.a.da.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.7	✓	18	5.3	odd	4
1690.2.b.f.339.12	yes	18	5.2	odd	4
1690.2.b.g.339.3	yes	18	65.12	odd	4
1690.2.b.g.339.16	yes	18	65.38	odd	4
1690.2.c.g.1689.8		18	65.47	even	4
1690.2.c.g.1689.11		18	65.18	even	4
1690.2.c.h.1689.8		18	65.57	even	4
1690.2.c.h.1689.11		18	65.8	even	4
8450.2.a.ct.1.7		9	13.12	even	2
8450.2.a.cw.1.3		9	5.4	even	2
8450.2.a.cx.1.7		9	1.1	even	1	trivial
8450.2.a.da.1.3		9	65.64	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.303663 q^{3} +1.00000 q^{4} +0.303663 q^{6} -1.08593 q^{7} +1.00000 q^{8} -2.90779 q^{9} -0.403887 q^{11} +0.303663 q^{12} -1.08593 q^{14} +1.00000 q^{16} +5.21387 q^{17} -2.90779 q^{18} +0.844510 q^{19} -0.329755 q^{21} -0.403887 q^{22} +1.76217 q^{23} +0.303663 q^{24} -1.79397 q^{27} -1.08593 q^{28} -5.96747 q^{29} -5.25338 q^{31} +1.00000 q^{32} -0.122645 q^{33} +5.21387 q^{34} -2.90779 q^{36} -4.42321 q^{37} +0.844510 q^{38} +3.20196 q^{41} -0.329755 q^{42} -11.4865 q^{43} -0.403887 q^{44} +1.76217 q^{46} +11.5048 q^{47} +0.303663 q^{48} -5.82076 q^{49} +1.58326 q^{51} -10.7632 q^{53} -1.79397 q^{54} -1.08593 q^{56} +0.256446 q^{57} -5.96747 q^{58} -7.28501 q^{59} -2.64135 q^{61} -5.25338 q^{62} +3.15764 q^{63} +1.00000 q^{64} -0.122645 q^{66} +10.5992 q^{67} +5.21387 q^{68} +0.535105 q^{69} -12.4325 q^{71} -2.90779 q^{72} -2.92837 q^{73} -4.42321 q^{74} +0.844510 q^{76} +0.438592 q^{77} +15.8880 q^{79} +8.17860 q^{81} +3.20196 q^{82} -12.6626 q^{83} -0.329755 q^{84} -11.4865 q^{86} -1.81210 q^{87} -0.403887 q^{88} +6.99770 q^{89} +1.76217 q^{92} -1.59525 q^{93} +11.5048 q^{94} +0.303663 q^{96} +9.47384 q^{97} -5.82076 q^{98} +1.17442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 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

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