LMFDB - Embedded newform 8450.2.a.cs.1.8

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 = [8,8,0,8,0,0,10,8,16,0,0,0,0,10,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 132x^{4} - 332x^{2} + 256 $ x^8 - 20x^6 + 132x^4 - 332*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.07108$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +3.07108 q^{3} +1.00000 q^{4} +3.07108 q^{6} +3.69510 q^{7} +1.00000 q^{8} +6.43154 q^{9} -1.06735 q^{11} +3.07108 q^{12} +3.69510 q^{14} +1.00000 q^{16} -2.79940 q^{17} +6.43154 q^{18} -2.19819 q^{19} +11.3479 q^{21} -1.06735 q^{22} -1.13083 q^{23} +3.07108 q^{24} +10.5385 q^{27} +3.69510 q^{28} -2.73644 q^{29} +4.20191 q^{31} +1.00000 q^{32} -3.27793 q^{33} -2.79940 q^{34} +6.43154 q^{36} +10.3193 q^{37} -2.19819 q^{38} -4.93821 q^{41} +11.3479 q^{42} -4.39637 q^{43} -1.06735 q^{44} -1.13083 q^{46} +10.5582 q^{47} +3.07108 q^{48} +6.65376 q^{49} -8.59720 q^{51} -3.81853 q^{53} +10.5385 q^{54} +3.69510 q^{56} -6.75081 q^{57} -2.73644 q^{58} -3.61033 q^{59} +15.2214 q^{61} +4.20191 q^{62} +23.7652 q^{63} +1.00000 q^{64} -3.27793 q^{66} +4.00000 q^{67} -2.79940 q^{68} -3.47288 q^{69} +12.2843 q^{71} +6.43154 q^{72} +1.23397 q^{73} +10.3193 q^{74} -2.19819 q^{76} -3.94398 q^{77} -14.2237 q^{79} +13.0701 q^{81} -4.93821 q^{82} -8.18235 q^{83} +11.3479 q^{84} -4.39637 q^{86} -8.40383 q^{87} -1.06735 q^{88} +8.80497 q^{89} -1.13083 q^{92} +12.9044 q^{93} +10.5582 q^{94} +3.07108 q^{96} -8.75081 q^{97} +6.65376 q^{98} -6.86472 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * q^98

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
$3$	3.07108	1.77309	0.886545	−	0.462643i	$-0.153099\pi$
$3$	3.07108	1.77309	0.886545	+	0.462643i	$0.153099\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.07108	1.25376
$7$	3.69510	1.39662	0.698308	−	0.715797i	$-0.253937\pi$
$7$	3.69510	1.39662	0.698308	+	0.715797i	$0.253937\pi$
$8$	1.00000	0.353553
$9$	6.43154	2.14385
$10$	0	0
$11$	−1.06735	−0.321819	−0.160910	−	0.986969i	$-0.551443\pi$
$11$	−1.06735	−0.321819	−0.160910	+	0.986969i	$0.551443\pi$
$12$	3.07108	0.886545
$13$	0	0
$13$	0	0
$14$	3.69510	0.987557
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.79940	−0.678955	−0.339478	−	0.940614i	$-0.610250\pi$
$17$	−2.79940	−0.678955	−0.339478	+	0.940614i	$0.610250\pi$
$18$	6.43154	1.51593
$19$	−2.19819	−0.504298	−0.252149	−	0.967688i	$-0.581137\pi$
$19$	−2.19819	−0.504298	−0.252149	+	0.967688i	$0.581137\pi$
$20$	0	0
$21$	11.3479	2.47633
$22$	−1.06735	−0.227560
$23$	−1.13083	−0.235795	−0.117897	−	0.993026i	$-0.537615\pi$
$23$	−1.13083	−0.235795	−0.117897	+	0.993026i	$0.537615\pi$
$24$	3.07108	0.626882
$25$	0	0
$26$	0	0
$27$	10.5385	2.02814
$28$	3.69510	0.698308
$29$	−2.73644	−0.508144	−0.254072	−	0.967185i	$-0.581770\pi$
$29$	−2.73644	−0.508144	−0.254072	+	0.967185i	$0.581770\pi$
$30$	0	0
$31$	4.20191	0.754686	0.377343	−	0.926074i	$-0.376838\pi$
$31$	4.20191	0.754686	0.377343	+	0.926074i	$0.376838\pi$
$32$	1.00000	0.176777
$33$	−3.27793	−0.570614
$34$	−2.79940	−0.480094
$35$	0	0
$36$	6.43154	1.07192
$37$	10.3193	1.69648	0.848239	−	0.529614i	$-0.177663\pi$
$37$	10.3193	1.69648	0.848239	+	0.529614i	$0.177663\pi$
$38$	−2.19819	−0.356593
$39$	0	0
$40$	0	0
$41$	−4.93821	−0.771220	−0.385610	−	0.922662i	$-0.626009\pi$
$41$	−4.93821	−0.771220	−0.385610	+	0.922662i	$0.626009\pi$
$42$	11.3479	1.75103
$43$	−4.39637	−0.670440	−0.335220	−	0.942140i	$-0.608811\pi$
$43$	−4.39637	−0.670440	−0.335220	+	0.942140i	$0.608811\pi$
$44$	−1.06735	−0.160910
$45$	0	0
$46$	−1.13083	−0.166732
$47$	10.5582	1.54007	0.770034	−	0.638003i	$-0.220239\pi$
$47$	10.5582	1.54007	0.770034	+	0.638003i	$0.220239\pi$
$48$	3.07108	0.443272
$49$	6.65376	0.950537
$50$	0	0
$51$	−8.59720	−1.20385
$52$	0	0
$53$	−3.81853	−0.524515	−0.262258	−	0.964998i	$-0.584467\pi$
$53$	−3.81853	−0.524515	−0.262258	+	0.964998i	$0.584467\pi$
$54$	10.5385	1.43411
$55$	0	0
$56$	3.69510	0.493778
$57$	−6.75081	−0.894166
$58$	−2.73644	−0.359312
$59$	−3.61033	−0.470025	−0.235013	−	0.971992i	$-0.575513\pi$
$59$	−3.61033	−0.470025	−0.235013	+	0.971992i	$0.575513\pi$
$60$	0	0
$61$	15.2214	1.94890	0.974448	−	0.224613i	$-0.0721117\pi$
$61$	15.2214	1.94890	0.974448	+	0.224613i	$0.0721117\pi$
$62$	4.20191	0.533644
$63$	23.7652	2.99413
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.27793	−0.403485
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.79940	−0.339478
$69$	−3.47288	−0.418086
$70$	0	0
$71$	12.2843	1.45788	0.728941	−	0.684577i	$-0.240013\pi$
$71$	12.2843	1.45788	0.728941	+	0.684577i	$0.240013\pi$
$72$	6.43154	0.757964
$73$	1.23397	0.144425	0.0722127	−	0.997389i	$-0.476994\pi$
$73$	1.23397	0.144425	0.0722127	+	0.997389i	$0.476994\pi$
$74$	10.3193	1.19959
$75$	0	0
$76$	−2.19819	−0.252149
$77$	−3.94398	−0.449458
$78$	0	0
$79$	−14.2237	−1.60029	−0.800145	−	0.599807i	$-0.795244\pi$
$79$	−14.2237	−1.60029	−0.800145	+	0.599807i	$0.795244\pi$
$80$	0	0
$81$	13.0701	1.45223
$82$	−4.93821	−0.545335
$83$	−8.18235	−0.898129	−0.449065	−	0.893499i	$-0.648243\pi$
$83$	−8.18235	−0.898129	−0.449065	+	0.893499i	$0.648243\pi$
$84$	11.3479	1.23816
$85$	0	0
$86$	−4.39637	−0.474073
$87$	−8.40383	−0.900985
$88$	−1.06735	−0.113780
$89$	8.80497	0.933325	0.466663	−	0.884435i	$-0.345456\pi$
$89$	8.80497	0.933325	0.466663	+	0.884435i	$0.345456\pi$
$90$	0	0
$91$	0	0
$92$	−1.13083	−0.117897
$93$	12.9044	1.33813
$94$	10.5582	1.08899
$95$	0	0
$96$	3.07108	0.313441
$97$	−8.75081	−0.888510	−0.444255	−	0.895900i	$-0.646532\pi$
$97$	−8.75081	−0.888510	−0.444255	+	0.895900i	$0.646532\pi$
$98$	6.65376	0.672131
$99$	−6.86472	−0.689931
$100$	0	0
$101$	−2.58283	−0.257001	−0.128501	−	0.991709i	$-0.541016\pi$
$101$	−2.58283	−0.257001	−0.128501	+	0.991709i	$0.541016\pi$
$102$	−8.59720	−0.851249
$103$	−3.00760	−0.296348	−0.148174	−	0.988961i	$-0.547340\pi$
$103$	−3.00760	−0.296348	−0.148174	+	0.988961i	$0.547340\pi$
$104$	0	0
$105$	0	0
$106$	−3.81853	−0.370888
$107$	−5.47595	−0.529380	−0.264690	−	0.964334i	$-0.585270\pi$
$107$	−5.47595	−0.529380	−0.264690	+	0.964334i	$0.585270\pi$
$108$	10.5385	1.01407
$109$	11.8872	1.13859	0.569294	−	0.822134i	$-0.307217\pi$
$109$	11.8872	1.13859	0.569294	+	0.822134i	$0.307217\pi$
$110$	0	0
$111$	31.6913	3.00801
$112$	3.69510	0.349154
$113$	9.26298	0.871388	0.435694	−	0.900095i	$-0.356503\pi$
$113$	9.26298	0.871388	0.435694	+	0.900095i	$0.356503\pi$
$114$	−6.75081	−0.632271
$115$	0	0
$116$	−2.73644	−0.254072
$117$	0	0
$118$	−3.61033	−0.332358
$119$	−10.3441	−0.948240
$120$	0	0
$121$	−9.86076	−0.896432
$122$	15.2214	1.37808
$123$	−15.1657	−1.36744
$124$	4.20191	0.377343
$125$	0	0
$126$	23.7652	2.11717
$127$	7.40397	0.656996	0.328498	−	0.944505i	$-0.393458\pi$
$127$	7.40397	0.656996	0.328498	+	0.944505i	$0.393458\pi$
$128$	1.00000	0.0883883
$129$	−13.5016	−1.18875
$130$	0	0
$131$	−7.35829	−0.642897	−0.321448	−	0.946927i	$-0.604170\pi$
$131$	−7.35829	−0.642897	−0.321448	+	0.946927i	$0.604170\pi$
$132$	−3.27793	−0.285307
$133$	−8.12252	−0.704311
$134$	4.00000	0.345547
$135$	0	0
$136$	−2.79940	−0.240047
$137$	−13.4046	−1.14523	−0.572615	−	0.819825i	$-0.694071\pi$
$137$	−13.4046	−1.14523	−0.572615	+	0.819825i	$0.694071\pi$
$138$	−3.47288	−0.295631
$139$	11.3583	0.963397	0.481699	−	0.876337i	$-0.340020\pi$
$139$	11.3583	0.963397	0.481699	+	0.876337i	$0.340020\pi$
$140$	0	0
$141$	32.4250	2.73068
$142$	12.2843	1.03088
$143$	0	0
$144$	6.43154	0.535962
$145$	0	0
$146$	1.23397	0.102124
$147$	20.4342	1.68539
$148$	10.3193	0.848239
$149$	−7.00132	−0.573570	−0.286785	−	0.957995i	$-0.592587\pi$
$149$	−7.00132	−0.573570	−0.286785	+	0.957995i	$0.592587\pi$
$150$	0	0
$151$	−0.194458	−0.0158248	−0.00791239	−	0.999969i	$-0.502519\pi$
$151$	−0.194458	−0.0158248	−0.00791239	+	0.999969i	$0.502519\pi$
$152$	−2.19819	−0.178296
$153$	−18.0045	−1.45558
$154$	−3.94398	−0.317815
$155$	0	0
$156$	0	0
$157$	−16.8889	−1.34788	−0.673940	−	0.738786i	$-0.735399\pi$
$157$	−16.8889	−1.34788	−0.673940	+	0.738786i	$0.735399\pi$
$158$	−14.2237	−1.13158
$159$	−11.7270	−0.930012
$160$	0	0
$161$	−4.17854	−0.329315
$162$	13.0701	1.02688
$163$	2.23165	0.174796	0.0873982	−	0.996173i	$-0.472145\pi$
$163$	2.23165	0.174796	0.0873982	+	0.996173i	$0.472145\pi$
$164$	−4.93821	−0.385610
$165$	0	0
$166$	−8.18235	−0.635073
$167$	2.03902	0.157784	0.0788920	−	0.996883i	$-0.474862\pi$
$167$	2.03902	0.157784	0.0788920	+	0.996883i	$0.474862\pi$
$168$	11.3479	0.875513
$169$	0	0
$170$	0	0
$171$	−14.1377	−1.08114
$172$	−4.39637	−0.335220
$173$	−10.8722	−0.826596	−0.413298	−	0.910596i	$-0.635623\pi$
$173$	−10.8722	−0.826596	−0.413298	+	0.910596i	$0.635623\pi$
$174$	−8.40383	−0.637093
$175$	0	0
$176$	−1.06735	−0.0804548
$177$	−11.0876	−0.833396
$178$	8.80497	0.659961
$179$	−3.91732	−0.292794	−0.146397	−	0.989226i	$-0.546768\pi$
$179$	−3.91732	−0.292794	−0.146397	+	0.989226i	$0.546768\pi$
$180$	0	0
$181$	14.1976	1.05530	0.527648	−	0.849463i	$-0.323074\pi$
$181$	14.1976	1.05530	0.527648	+	0.849463i	$0.323074\pi$
$182$	0	0
$183$	46.7460	3.45557
$184$	−1.13083	−0.0833661
$185$	0	0
$186$	12.9044	0.946198
$187$	2.98795	0.218501
$188$	10.5582	0.770034
$189$	38.9409	2.83254
$190$	0	0
$191$	−0.336811	−0.0243708	−0.0121854	−	0.999926i	$-0.503879\pi$
$191$	−0.336811	−0.0243708	−0.0121854	+	0.999926i	$0.503879\pi$
$192$	3.07108	0.221636
$193$	−6.76518	−0.486968	−0.243484	−	0.969905i	$-0.578290\pi$
$193$	−6.76518	−0.486968	−0.243484	+	0.969905i	$0.578290\pi$
$194$	−8.75081	−0.628271
$195$	0	0
$196$	6.65376	0.475269
$197$	−14.9363	−1.06417	−0.532085	−	0.846691i	$-0.678591\pi$
$197$	−14.9363	−1.06417	−0.532085	+	0.846691i	$0.678591\pi$
$198$	−6.86472	−0.487855
$199$	−0.302580	−0.0214493	−0.0107247	−	0.999942i	$-0.503414\pi$
$199$	−0.302580	−0.0214493	−0.0107247	+	0.999942i	$0.503414\pi$
$200$	0	0
$201$	12.2843	0.866469
$202$	−2.58283	−0.181727
$203$	−10.1114	−0.709682
$204$	−8.59720	−0.601924
$205$	0	0
$206$	−3.00760	−0.209550
$207$	−7.27299	−0.505508
$208$	0	0
$209$	2.34624	0.162293
$210$	0	0
$211$	0.404566	0.0278515	0.0139258	−	0.999903i	$-0.495567\pi$
$211$	0.404566	0.0278515	0.0139258	+	0.999903i	$0.495567\pi$
$212$	−3.81853	−0.262258
$213$	37.7262	2.58495
$214$	−5.47595	−0.374328
$215$	0	0
$216$	10.5385	0.717056
$217$	15.5265	1.05401
$218$	11.8872	0.805103
$219$	3.78962	0.256079
$220$	0	0
$221$	0	0
$222$	31.6913	2.12698
$223$	0.832022	0.0557163	0.0278581	−	0.999612i	$-0.491131\pi$
$223$	0.832022	0.0557163	0.0278581	+	0.999612i	$0.491131\pi$
$224$	3.69510	0.246889
$225$	0	0
$226$	9.26298	0.616164
$227$	−7.77546	−0.516075	−0.258038	−	0.966135i	$-0.583076\pi$
$227$	−7.77546	−0.516075	−0.258038	+	0.966135i	$0.583076\pi$
$228$	−6.75081	−0.447083
$229$	7.73762	0.511316	0.255658	−	0.966767i	$-0.417708\pi$
$229$	7.73762	0.511316	0.255658	+	0.966767i	$0.417708\pi$
$230$	0	0
$231$	−12.1123	−0.796929
$232$	−2.73644	−0.179656
$233$	−6.86070	−0.449460	−0.224730	−	0.974421i	$-0.572150\pi$
$233$	−6.86070	−0.449460	−0.224730	+	0.974421i	$0.572150\pi$
$234$	0	0
$235$	0	0
$236$	−3.61033	−0.235013
$237$	−43.6821	−2.83746
$238$	−10.3441	−0.670507
$239$	−22.3277	−1.44426	−0.722130	−	0.691757i	$-0.756837\pi$
$239$	−22.3277	−1.44426	−0.722130	+	0.691757i	$0.756837\pi$
$240$	0	0
$241$	−5.68717	−0.366343	−0.183172	−	0.983081i	$-0.558636\pi$
$241$	−5.68717	−0.366343	−0.183172	+	0.983081i	$0.558636\pi$
$242$	−9.86076	−0.633873
$243$	8.52366	0.546793
$244$	15.2214	0.974448
$245$	0	0
$246$	−15.1657	−0.966927
$247$	0	0
$248$	4.20191	0.266822
$249$	−25.1286	−1.59246
$250$	0	0
$251$	−17.6162	−1.11193	−0.555963	−	0.831207i	$-0.687650\pi$
$251$	−17.6162	−1.11193	−0.555963	+	0.831207i	$0.687650\pi$
$252$	23.7652	1.49707
$253$	1.20700	0.0758833
$254$	7.40397	0.464567
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.8167	−1.61040	−0.805201	−	0.593001i	$-0.797943\pi$
$257$	−25.8167	−1.61040	−0.805201	+	0.593001i	$0.797943\pi$
$258$	−13.5016	−0.840574
$259$	38.1307	2.36933
$260$	0	0
$261$	−17.5995	−1.08938
$262$	−7.35829	−0.454597
$263$	−21.2163	−1.30825	−0.654125	−	0.756386i	$-0.726963\pi$
$263$	−21.2163	−1.30825	−0.654125	+	0.756386i	$0.726963\pi$
$264$	−3.27793	−0.201743
$265$	0	0
$266$	−8.12252	−0.498023
$267$	27.0408	1.65487
$268$	4.00000	0.244339
$269$	14.4896	0.883445	0.441722	−	0.897152i	$-0.354368\pi$
$269$	14.4896	0.883445	0.441722	+	0.897152i	$0.354368\pi$
$270$	0	0
$271$	−29.0838	−1.76671	−0.883357	−	0.468701i	$-0.844722\pi$
$271$	−29.0838	−1.76671	−0.883357	+	0.468701i	$0.844722\pi$
$272$	−2.79940	−0.169739
$273$	0	0
$274$	−13.4046	−0.809799
$275$	0	0
$276$	−3.47288	−0.209043
$277$	−2.54147	−0.152702	−0.0763510	−	0.997081i	$-0.524327\pi$
$277$	−2.54147	−0.152702	−0.0763510	+	0.997081i	$0.524327\pi$
$278$	11.3583	0.681225
$279$	27.0248	1.61793
$280$	0	0
$281$	15.2066	0.907149	0.453574	−	0.891218i	$-0.350149\pi$
$281$	15.2066	0.907149	0.453574	+	0.891218i	$0.350149\pi$
$282$	32.4250	1.93088
$283$	−24.7921	−1.47374	−0.736868	−	0.676037i	$-0.763696\pi$
$283$	−24.7921	−1.47374	−0.736868	+	0.676037i	$0.763696\pi$
$284$	12.2843	0.728941
$285$	0	0
$286$	0	0
$287$	−18.2472	−1.07710
$288$	6.43154	0.378982
$289$	−9.16334	−0.539020
$290$	0	0
$291$	−26.8744	−1.57541
$292$	1.23397	0.0722127
$293$	21.0630	1.23051	0.615256	−	0.788328i	$-0.289053\pi$
$293$	21.0630	1.23051	0.615256	+	0.788328i	$0.289053\pi$
$294$	20.4342	1.19175
$295$	0	0
$296$	10.3193	0.599795
$297$	−11.2483	−0.652695
$298$	−7.00132	−0.405575
$299$	0	0
$300$	0	0
$301$	−16.2450	−0.936348
$302$	−0.194458	−0.0111898
$303$	−7.93208	−0.455686
$304$	−2.19819	−0.126075
$305$	0	0
$306$	−18.0045	−1.02925
$307$	−8.89181	−0.507483	−0.253741	−	0.967272i	$-0.581661\pi$
$307$	−8.89181	−0.507483	−0.253741	+	0.967272i	$0.581661\pi$
$308$	−3.94398	−0.224729
$309$	−9.23659	−0.525451
$310$	0	0
$311$	−28.7015	−1.62751	−0.813756	−	0.581206i	$-0.802581\pi$
$311$	−28.7015	−1.62751	−0.813756	+	0.581206i	$0.802581\pi$
$312$	0	0
$313$	7.84120	0.443211	0.221606	−	0.975136i	$-0.428870\pi$
$313$	7.84120	0.443211	0.221606	+	0.975136i	$0.428870\pi$
$314$	−16.8889	−0.953095
$315$	0	0
$316$	−14.2237	−0.800145
$317$	−6.17741	−0.346958	−0.173479	−	0.984838i	$-0.555501\pi$
$317$	−6.17741	−0.346958	−0.173479	+	0.984838i	$0.555501\pi$
$318$	−11.7270	−0.657618
$319$	2.92075	0.163530
$320$	0	0
$321$	−16.8171	−0.938639
$322$	−4.17854	−0.232861
$323$	6.15361	0.342396
$324$	13.0701	0.726115
$325$	0	0
$326$	2.23165	0.123600
$327$	36.5066	2.01882
$328$	−4.93821	−0.272667
$329$	39.0135	2.15088
$330$	0	0
$331$	26.6566	1.46518	0.732590	−	0.680670i	$-0.238311\pi$
$331$	26.6566	1.46518	0.732590	+	0.680670i	$0.238311\pi$
$332$	−8.18235	−0.449065
$333$	66.3688	3.63699
$334$	2.03902	0.111570
$335$	0	0
$336$	11.3479	0.619081
$337$	−22.4283	−1.22175	−0.610874	−	0.791728i	$-0.709182\pi$
$337$	−22.4283	−1.22175	−0.610874	+	0.791728i	$0.709182\pi$
$338$	0	0
$339$	28.4474	1.54505
$340$	0	0
$341$	−4.48493	−0.242872
$342$	−14.1377	−0.764480
$343$	−1.27939	−0.0690808
$344$	−4.39637	−0.237036
$345$	0	0
$346$	−10.8722	−0.584492
$347$	−0.412291	−0.0221329	−0.0110665	−	0.999939i	$-0.503523\pi$
$347$	−0.412291	−0.0221329	−0.0110665	+	0.999939i	$0.503523\pi$
$348$	−8.40383	−0.450492
$349$	−3.34125	−0.178853	−0.0894264	−	0.995993i	$-0.528503\pi$
$349$	−3.34125	−0.178853	−0.0894264	+	0.995993i	$0.528503\pi$
$350$	0	0
$351$	0	0
$352$	−1.06735	−0.0568901
$353$	21.1208	1.12415	0.562075	−	0.827087i	$-0.310003\pi$
$353$	21.1208	1.12415	0.562075	+	0.827087i	$0.310003\pi$
$354$	−11.0876	−0.589300
$355$	0	0
$356$	8.80497	0.466663
$357$	−31.7675	−1.68131
$358$	−3.91732	−0.207037
$359$	−5.57954	−0.294477	−0.147238	−	0.989101i	$-0.547038\pi$
$359$	−5.57954	−0.294477	−0.147238	+	0.989101i	$0.547038\pi$
$360$	0	0
$361$	−14.1680	−0.745683
$362$	14.1976	0.746208
$363$	−30.2832	−1.58945
$364$	0	0
$365$	0	0
$366$	46.7460	2.44346
$367$	1.30192	0.0679595	0.0339797	−	0.999423i	$-0.489182\pi$
$367$	1.30192	0.0679595	0.0339797	+	0.999423i	$0.489182\pi$
$368$	−1.13083	−0.0589487
$369$	−31.7603	−1.65338
$370$	0	0
$371$	−14.1098	−0.732546
$372$	12.9044	0.669063
$373$	−33.9971	−1.76030	−0.880152	−	0.474691i	$-0.842560\pi$
$373$	−33.9971	−1.76030	−0.880152	+	0.474691i	$0.842560\pi$
$374$	2.98795	0.154503
$375$	0	0
$376$	10.5582	0.544496
$377$	0	0
$378$	38.9409	2.00291
$379$	4.67768	0.240276	0.120138	−	0.992757i	$-0.461666\pi$
$379$	4.67768	0.240276	0.120138	+	0.992757i	$0.461666\pi$
$380$	0	0
$381$	22.7382	1.16491
$382$	−0.336811	−0.0172328
$383$	−13.7388	−0.702018	−0.351009	−	0.936372i	$-0.614161\pi$
$383$	−13.7388	−0.702018	−0.351009	+	0.936372i	$0.614161\pi$
$384$	3.07108	0.156720
$385$	0	0
$386$	−6.76518	−0.344338
$387$	−28.2754	−1.43732
$388$	−8.75081	−0.444255
$389$	17.9138	0.908268	0.454134	−	0.890933i	$-0.349949\pi$
$389$	17.9138	0.908268	0.454134	+	0.890933i	$0.349949\pi$
$390$	0	0
$391$	3.16566	0.160094
$392$	6.65376	0.336066
$393$	−22.5979	−1.13991
$394$	−14.9363	−0.752481
$395$	0	0
$396$	−6.86472	−0.344965
$397$	0.458511	0.0230120	0.0115060	−	0.999934i	$-0.496337\pi$
$397$	0.458511	0.0230120	0.0115060	+	0.999934i	$0.496337\pi$
$398$	−0.302580	−0.0151670
$399$	−24.9449	−1.24881
$400$	0	0
$401$	26.0554	1.30114	0.650572	−	0.759445i	$-0.274529\pi$
$401$	26.0554	1.30114	0.650572	+	0.759445i	$0.274529\pi$
$402$	12.2843	0.612686
$403$	0	0
$404$	−2.58283	−0.128501
$405$	0	0
$406$	−10.1114	−0.501821
$407$	−11.0143	−0.545959
$408$	−8.59720	−0.425625
$409$	−0.358531	−0.0177282	−0.00886411	−	0.999961i	$-0.502822\pi$
$409$	−0.358531	−0.0177282	−0.00886411	+	0.999961i	$0.502822\pi$
$410$	0	0
$411$	−41.1665	−2.03059
$412$	−3.00760	−0.148174
$413$	−13.3405	−0.656445
$414$	−7.27299	−0.357448
$415$	0	0
$416$	0	0
$417$	34.8822	1.70819
$418$	2.34624	0.114758
$419$	−19.3184	−0.943766	−0.471883	−	0.881661i	$-0.656426\pi$
$419$	−19.3184	−0.943766	−0.471883	+	0.881661i	$0.656426\pi$
$420$	0	0
$421$	−0.220426	−0.0107429	−0.00537144	−	0.999986i	$-0.501710\pi$
$421$	−0.220426	−0.0107429	−0.00537144	+	0.999986i	$0.501710\pi$
$422$	0.404566	0.0196940
$423$	67.9053	3.30167
$424$	−3.81853	−0.185444
$425$	0	0
$426$	37.7262	1.82784
$427$	56.2445	2.72186
$428$	−5.47595	−0.264690
$429$	0	0
$430$	0	0
$431$	19.9696	0.961902	0.480951	−	0.876747i	$-0.340291\pi$
$431$	19.9696	0.961902	0.480951	+	0.876747i	$0.340291\pi$
$432$	10.5385	0.507035
$433$	31.4155	1.50973	0.754867	−	0.655878i	$-0.227701\pi$
$433$	31.4155	1.50973	0.754867	+	0.655878i	$0.227701\pi$
$434$	15.5265	0.745295
$435$	0	0
$436$	11.8872	0.569294
$437$	2.48578	0.118911
$438$	3.78962	0.181075
$439$	−25.6139	−1.22248	−0.611242	−	0.791444i	$-0.709330\pi$
$439$	−25.6139	−1.22248	−0.611242	+	0.791444i	$0.709330\pi$
$440$	0	0
$441$	42.7939	2.03781
$442$	0	0
$443$	28.4024	1.34944	0.674719	−	0.738074i	$-0.264265\pi$
$443$	28.4024	1.34944	0.674719	+	0.738074i	$0.264265\pi$
$444$	31.6913	1.50400
$445$	0	0
$446$	0.832022	0.0393974
$447$	−21.5016	−1.01699
$448$	3.69510	0.174577
$449$	−22.4217	−1.05815	−0.529073	−	0.848576i	$-0.677460\pi$
$449$	−22.4217	−1.05815	−0.529073	+	0.848576i	$0.677460\pi$
$450$	0	0
$451$	5.27082	0.248193
$452$	9.26298	0.435694
$453$	−0.597197	−0.0280587
$454$	−7.77546	−0.364920
$455$	0	0
$456$	−6.75081	−0.316136
$457$	23.5987	1.10390	0.551949	−	0.833878i	$-0.313884\pi$
$457$	23.5987	1.10390	0.551949	+	0.833878i	$0.313884\pi$
$458$	7.73762	0.361555
$459$	−29.5016	−1.37702
$460$	0	0
$461$	−32.5063	−1.51397	−0.756986	−	0.653432i	$-0.773329\pi$
$461$	−32.5063	−1.51397	−0.756986	+	0.653432i	$0.773329\pi$
$462$	−12.1123	−0.563514
$463$	−7.70367	−0.358020	−0.179010	−	0.983847i	$-0.557289\pi$
$463$	−7.70367	−0.358020	−0.179010	+	0.983847i	$0.557289\pi$
$464$	−2.73644	−0.127036
$465$	0	0
$466$	−6.86070	−0.317816
$467$	−20.9339	−0.968704	−0.484352	−	0.874873i	$-0.660945\pi$
$467$	−20.9339	−0.968704	−0.484352	+	0.874873i	$0.660945\pi$
$468$	0	0
$469$	14.7804	0.682495
$470$	0	0
$471$	−51.8672	−2.38991
$472$	−3.61033	−0.166179
$473$	4.69248	0.215761
$474$	−43.6821	−2.00639
$475$	0	0
$476$	−10.3441	−0.474120
$477$	−24.5590	−1.12448
$478$	−22.3277	−1.02125
$479$	0.980499	0.0448002	0.0224001	−	0.999749i	$-0.492869\pi$
$479$	0.980499	0.0448002	0.0224001	+	0.999749i	$0.492869\pi$
$480$	0	0
$481$	0	0
$482$	−5.68717	−0.259044
$483$	−12.8326	−0.583905
$484$	−9.86076	−0.448216
$485$	0	0
$486$	8.52366	0.386641
$487$	38.6283	1.75041	0.875207	−	0.483749i	$-0.160725\pi$
$487$	38.6283	1.75041	0.875207	+	0.483749i	$0.160725\pi$
$488$	15.2214	0.689039
$489$	6.85358	0.309929
$490$	0	0
$491$	−17.7068	−0.799099	−0.399549	−	0.916712i	$-0.630833\pi$
$491$	−17.7068	−0.799099	−0.399549	+	0.916712i	$0.630833\pi$
$492$	−15.1657	−0.683721
$493$	7.66040	0.345007
$494$	0	0
$495$	0	0
$496$	4.20191	0.188672
$497$	45.3918	2.03610
$498$	−25.1286	−1.12604
$499$	35.5402	1.59100	0.795499	−	0.605954i	$-0.207209\pi$
$499$	35.5402	1.59100	0.795499	+	0.605954i	$0.207209\pi$
$500$	0	0
$501$	6.26199	0.279765
$502$	−17.6162	−0.786250
$503$	44.3839	1.97898	0.989490	−	0.144600i	$-0.0461894\pi$
$503$	44.3839	1.97898	0.989490	+	0.144600i	$0.0461894\pi$
$504$	23.7652	1.05859
$505$	0	0
$506$	1.20700	0.0536576
$507$	0	0
$508$	7.40397	0.328498
$509$	0.715949	0.0317339	0.0158669	−	0.999874i	$-0.494949\pi$
$509$	0.715949	0.0317339	0.0158669	+	0.999874i	$0.494949\pi$
$510$	0	0
$511$	4.55964	0.201707
$512$	1.00000	0.0441942
$513$	−23.1657	−1.02279
$514$	−25.8167	−1.13873
$515$	0	0
$516$	−13.5016	−0.594375
$517$	−11.2693	−0.495623
$518$	38.1307	1.67537
$519$	−33.3893	−1.46563
$520$	0	0
$521$	7.33132	0.321191	0.160595	−	0.987020i	$-0.448659\pi$
$521$	7.33132	0.321191	0.160595	+	0.987020i	$0.448659\pi$
$522$	−17.5995	−0.770310
$523$	11.8872	0.519791	0.259895	−	0.965637i	$-0.416312\pi$
$523$	11.8872	0.519791	0.259895	+	0.965637i	$0.416312\pi$
$524$	−7.35829	−0.321448
$525$	0	0
$526$	−21.2163	−0.925073
$527$	−11.7629	−0.512398
$528$	−3.27793	−0.142654
$529$	−21.7212	−0.944401
$530$	0	0
$531$	−23.2200	−1.00766
$532$	−8.12252	−0.352156
$533$	0	0
$534$	27.0408	1.17017
$535$	0	0
$536$	4.00000	0.172774
$537$	−12.0304	−0.519150
$538$	14.4896	0.624690
$539$	−7.10191	−0.305901
$540$	0	0
$541$	−20.5115	−0.881856	−0.440928	−	0.897542i	$-0.645351\pi$
$541$	−20.5115	−0.881856	−0.440928	+	0.897542i	$0.645351\pi$
$542$	−29.0838	−1.24925
$543$	43.6019	1.87114
$544$	−2.79940	−0.120023
$545$	0	0
$546$	0	0
$547$	16.4268	0.702358	0.351179	−	0.936308i	$-0.385781\pi$
$547$	16.4268	0.702358	0.351179	+	0.936308i	$0.385781\pi$
$548$	−13.4046	−0.572615
$549$	97.8968	4.17813
$550$	0	0
$551$	6.01520	0.256256
$552$	−3.47288	−0.147816
$553$	−52.5579	−2.23499
$554$	−2.54147	−0.107977
$555$	0	0
$556$	11.3583	0.481699
$557$	−36.1459	−1.53155	−0.765776	−	0.643107i	$-0.777645\pi$
$557$	−36.1459	−1.53155	−0.765776	+	0.643107i	$0.777645\pi$
$558$	27.0248	1.14405
$559$	0	0
$560$	0	0
$561$	9.17625	0.387421
$562$	15.2066	0.641451
$563$	26.5601	1.11938	0.559688	−	0.828703i	$-0.310921\pi$
$563$	26.5601	1.11938	0.559688	+	0.828703i	$0.310921\pi$
$564$	32.4250	1.36534
$565$	0	0
$566$	−24.7921	−1.04209
$567$	48.2952	2.02821
$568$	12.2843	0.515439
$569$	43.1841	1.81037	0.905186	−	0.425016i	$-0.139731\pi$
$569$	43.1841	1.81037	0.905186	+	0.425016i	$0.139731\pi$
$570$	0	0
$571$	31.5110	1.31870	0.659348	−	0.751838i	$-0.270832\pi$
$571$	31.5110	1.31870	0.659348	+	0.751838i	$0.270832\pi$
$572$	0	0
$573$	−1.03437	−0.0432116
$574$	−18.2472	−0.761623
$575$	0	0
$576$	6.43154	0.267981
$577$	26.9555	1.12217	0.561086	−	0.827758i	$-0.310384\pi$
$577$	26.9555	1.12217	0.561086	+	0.827758i	$0.310384\pi$
$578$	−9.16334	−0.381145
$579$	−20.7764	−0.863438
$580$	0	0
$581$	−30.2346	−1.25434
$582$	−26.8744	−1.11398
$583$	4.07572	0.168799
$584$	1.23397	0.0510621
$585$	0	0
$586$	21.0630	0.870103
$587$	29.7262	1.22693	0.613465	−	0.789722i	$-0.289775\pi$
$587$	29.7262	1.22693	0.613465	+	0.789722i	$0.289775\pi$
$588$	20.4342	0.842694
$589$	−9.23659	−0.380587
$590$	0	0
$591$	−45.8707	−1.88687
$592$	10.3193	0.424119
$593$	−15.3561	−0.630600	−0.315300	−	0.948992i	$-0.602105\pi$
$593$	−15.3561	−0.630600	−0.315300	+	0.948992i	$0.602105\pi$
$594$	−11.2483	−0.461525
$595$	0	0
$596$	−7.00132	−0.286785
$597$	−0.929248	−0.0380316
$598$	0	0
$599$	−23.6928	−0.968061	−0.484030	−	0.875051i	$-0.660828\pi$
$599$	−23.6928	−0.968061	−0.484030	+	0.875051i	$0.660828\pi$
$600$	0	0
$601$	−3.49668	−0.142632	−0.0713162	−	0.997454i	$-0.522720\pi$
$601$	−3.49668	−0.142632	−0.0713162	+	0.997454i	$0.522720\pi$
$602$	−16.2450	−0.662098
$603$	25.7262	1.04765
$604$	−0.194458	−0.00791239
$605$	0	0
$606$	−7.93208	−0.322219
$607$	37.2982	1.51389	0.756944	−	0.653479i	$-0.226691\pi$
$607$	37.2982	1.51389	0.756944	+	0.653479i	$0.226691\pi$
$608$	−2.19819	−0.0891482
$609$	−31.0530	−1.25833
$610$	0	0
$611$	0	0
$612$	−18.0045	−0.727788
$613$	15.3242	0.618939	0.309469	−	0.950909i	$-0.399849\pi$
$613$	15.3242	0.618939	0.309469	+	0.950909i	$0.399849\pi$
$614$	−8.89181	−0.358844
$615$	0	0
$616$	−3.94398	−0.158907
$617$	5.07240	0.204207	0.102103	−	0.994774i	$-0.467443\pi$
$617$	5.07240	0.204207	0.102103	+	0.994774i	$0.467443\pi$
$618$	−9.23659	−0.371550
$619$	−20.6770	−0.831079	−0.415540	−	0.909575i	$-0.636407\pi$
$619$	−20.6770	−0.831079	−0.415540	+	0.909575i	$0.636407\pi$
$620$	0	0
$621$	−11.9173	−0.478226
$622$	−28.7015	−1.15083
$623$	32.5352	1.30350
$624$	0	0
$625$	0	0
$626$	7.84120	0.313398
$627$	7.20550	0.287760
$628$	−16.8889	−0.673940
$629$	−28.8878	−1.15183
$630$	0	0
$631$	21.4904	0.855521	0.427760	−	0.903892i	$-0.359303\pi$
$631$	21.4904	0.855521	0.427760	+	0.903892i	$0.359303\pi$
$632$	−14.2237	−0.565788
$633$	1.24246	0.0493832
$634$	−6.17741	−0.245336
$635$	0	0
$636$	−11.7270	−0.465006
$637$	0	0
$638$	2.92075	0.115634
$639$	79.0071	3.12547
$640$	0	0
$641$	35.7724	1.41293	0.706463	−	0.707750i	$-0.250290\pi$
$641$	35.7724	1.41293	0.706463	+	0.707750i	$0.250290\pi$
$642$	−16.8171	−0.663718
$643$	28.0216	1.10507	0.552533	−	0.833491i	$-0.313662\pi$
$643$	28.0216	1.10507	0.552533	+	0.833491i	$0.313662\pi$
$644$	−4.17854	−0.164658
$645$	0	0
$646$	6.15361	0.242111
$647$	−19.4623	−0.765140	−0.382570	−	0.923926i	$-0.624961\pi$
$647$	−19.4623	−0.765140	−0.382570	+	0.923926i	$0.624961\pi$
$648$	13.0701	0.513441
$649$	3.85350	0.151263
$650$	0	0
$651$	47.6831	1.86885
$652$	2.23165	0.0873982
$653$	12.9628	0.507272	0.253636	−	0.967300i	$-0.418373\pi$
$653$	12.9628	0.507272	0.253636	+	0.967300i	$0.418373\pi$
$654$	36.5066	1.42752
$655$	0	0
$656$	−4.93821	−0.192805
$657$	7.93633	0.309626
$658$	39.0135	1.52091
$659$	2.20670	0.0859609	0.0429804	−	0.999076i	$-0.486315\pi$
$659$	2.20670	0.0859609	0.0429804	+	0.999076i	$0.486315\pi$
$660$	0	0
$661$	−43.5415	−1.69357	−0.846784	−	0.531937i	$-0.821464\pi$
$661$	−43.5415	−1.69357	−0.846784	+	0.531937i	$0.821464\pi$
$662$	26.6566	1.03604
$663$	0	0
$664$	−8.18235	−0.317537
$665$	0	0
$666$	66.3688	2.57174
$667$	3.09446	0.119818
$668$	2.03902	0.0788920
$669$	2.55521	0.0987900
$670$	0	0
$671$	−16.2466	−0.627192
$672$	11.3479	0.437757
$673$	0.588990	0.0227039	0.0113519	−	0.999936i	$-0.496386\pi$
$673$	0.588990	0.0227039	0.0113519	+	0.999936i	$0.496386\pi$
$674$	−22.4283	−0.863907
$675$	0	0
$676$	0	0
$677$	−7.61779	−0.292775	−0.146388	−	0.989227i	$-0.546765\pi$
$677$	−7.61779	−0.292775	−0.146388	+	0.989227i	$0.546765\pi$
$678$	28.4474	1.09251
$679$	−32.3351	−1.24091
$680$	0	0
$681$	−23.8791	−0.915048
$682$	−4.48493	−0.171737
$683$	28.3647	1.08534	0.542672	−	0.839944i	$-0.317413\pi$
$683$	28.3647	1.08534	0.542672	+	0.839944i	$0.317413\pi$
$684$	−14.1377	−0.540569
$685$	0	0
$686$	−1.27939	−0.0488475
$687$	23.7629	0.906609
$688$	−4.39637	−0.167610
$689$	0	0
$690$	0	0
$691$	−41.3046	−1.57130	−0.785651	−	0.618670i	$-0.787672\pi$
$691$	−41.3046	−1.57130	−0.785651	+	0.618670i	$0.787672\pi$
$692$	−10.8722	−0.413298
$693$	−25.3658	−0.963568
$694$	−0.412291	−0.0156503
$695$	0	0
$696$	−8.40383	−0.318546
$697$	13.8241	0.523624
$698$	−3.34125	−0.126468
$699$	−21.0698	−0.796932
$700$	0	0
$701$	32.7428	1.23668	0.618340	−	0.785911i	$-0.287805\pi$
$701$	32.7428	1.23668	0.618340	+	0.785911i	$0.287805\pi$
$702$	0	0
$703$	−22.6837	−0.855531
$704$	−1.06735	−0.0402274
$705$	0	0
$706$	21.1208	0.794894
$707$	−9.54381	−0.358932
$708$	−11.0876	−0.416698
$709$	−21.5484	−0.809267	−0.404633	−	0.914479i	$-0.632601\pi$
$709$	−21.5484	−0.809267	−0.404633	+	0.914479i	$0.632601\pi$
$710$	0	0
$711$	−91.4802	−3.43078
$712$	8.80497	0.329980
$713$	−4.75166	−0.177951
$714$	−31.7675	−1.18887
$715$	0	0
$716$	−3.91732	−0.146397
$717$	−68.5703	−2.56080
$718$	−5.57954	−0.208227
$719$	42.0862	1.56955	0.784775	−	0.619780i	$-0.212778\pi$
$719$	42.0862	1.56955	0.784775	+	0.619780i	$0.212778\pi$
$720$	0	0
$721$	−11.1134	−0.413884
$722$	−14.1680	−0.527278
$723$	−17.4658	−0.649559
$724$	14.1976	0.527648
$725$	0	0
$726$	−30.2832	−1.12391
$727$	21.7707	0.807429	0.403715	−	0.914885i	$-0.367719\pi$
$727$	21.7707	0.807429	0.403715	+	0.914885i	$0.367719\pi$
$728$	0	0
$729$	−13.0334	−0.482718
$730$	0	0
$731$	12.3072	0.455199
$732$	46.7460	1.72778
$733$	16.2319	0.599541	0.299770	−	0.954011i	$-0.403090\pi$
$733$	16.2319	0.599541	0.299770	+	0.954011i	$0.403090\pi$
$734$	1.30192	0.0480546
$735$	0	0
$736$	−1.13083	−0.0416830
$737$	−4.26941	−0.157266
$738$	−31.7603	−1.16911
$739$	−18.4432	−0.678445	−0.339222	−	0.940706i	$-0.610164\pi$
$739$	−18.4432	−0.678445	−0.339222	+	0.940706i	$0.610164\pi$
$740$	0	0
$741$	0	0
$742$	−14.1098	−0.517989
$743$	8.21194	0.301267	0.150633	−	0.988590i	$-0.451869\pi$
$743$	8.21194	0.301267	0.150633	+	0.988590i	$0.451869\pi$
$744$	12.9044	0.473099
$745$	0	0
$746$	−33.9971	−1.24472
$747$	−52.6251	−1.92545
$748$	2.98795	0.109250
$749$	−20.2342	−0.739341
$750$	0	0
$751$	38.0908	1.38995	0.694977	−	0.719032i	$-0.255414\pi$
$751$	38.0908	1.38995	0.694977	+	0.719032i	$0.255414\pi$
$752$	10.5582	0.385017
$753$	−54.1008	−1.97154
$754$	0	0
$755$	0	0
$756$	38.9409	1.41627
$757$	−25.2668	−0.918336	−0.459168	−	0.888350i	$-0.651852\pi$
$757$	−25.2668	−0.918336	−0.459168	+	0.888350i	$0.651852\pi$
$758$	4.67768	0.169901
$759$	3.70679	0.134548
$760$	0	0
$761$	−36.5040	−1.32327	−0.661634	−	0.749827i	$-0.730137\pi$
$761$	−36.5040	−1.32327	−0.661634	+	0.749827i	$0.730137\pi$
$762$	22.7382	0.823718
$763$	43.9244	1.59017
$764$	−0.336811	−0.0121854
$765$	0	0
$766$	−13.7388	−0.496402
$767$	0	0
$768$	3.07108	0.110818
$769$	−27.7956	−1.00234	−0.501168	−	0.865350i	$-0.667096\pi$
$769$	−27.7956	−1.00234	−0.501168	+	0.865350i	$0.667096\pi$
$770$	0	0
$771$	−79.2852	−2.85539
$772$	−6.76518	−0.243484
$773$	3.09705	0.111393	0.0556965	−	0.998448i	$-0.482262\pi$
$773$	3.09705	0.111393	0.0556965	+	0.998448i	$0.482262\pi$
$774$	−28.2754	−1.01634
$775$	0	0
$776$	−8.75081	−0.314136
$777$	117.103	4.20103
$778$	17.9138	0.642243
$779$	10.8551	0.388925
$780$	0	0
$781$	−13.1117	−0.469174
$782$	3.16566	0.113204
$783$	−28.8381	−1.03059
$784$	6.65376	0.237634
$785$	0	0
$786$	−22.5979	−0.806040
$787$	30.4310	1.08475	0.542374	−	0.840137i	$-0.317526\pi$
$787$	30.4310	1.08475	0.542374	+	0.840137i	$0.317526\pi$
$788$	−14.9363	−0.532085
$789$	−65.1568	−2.31964
$790$	0	0
$791$	34.2276	1.21699
$792$	−6.86472	−0.243927
$793$	0	0
$794$	0.458511	0.0162720
$795$	0	0
$796$	−0.302580	−0.0107247
$797$	47.5544	1.68446	0.842231	−	0.539116i	$-0.181242\pi$
$797$	47.5544	1.68446	0.842231	+	0.539116i	$0.181242\pi$
$798$	−24.9449	−0.883040
$799$	−29.5566	−1.04564
$800$	0	0
$801$	56.6295	2.00091
$802$	26.0554	0.920048
$803$	−1.31708	−0.0464788
$804$	12.2843	0.433235
$805$	0	0
$806$	0	0
$807$	44.4986	1.56643
$808$	−2.58283	−0.0908636
$809$	−36.0836	−1.26863	−0.634316	−	0.773074i	$-0.718718\pi$
$809$	−36.0836	−1.26863	−0.634316	+	0.773074i	$0.718718\pi$
$810$	0	0
$811$	36.3456	1.27627	0.638134	−	0.769926i	$-0.279707\pi$
$811$	36.3456	1.27627	0.638134	+	0.769926i	$0.279707\pi$
$812$	−10.1114	−0.354841
$813$	−89.3186	−3.13254
$814$	−11.0143	−0.386051
$815$	0	0
$816$	−8.59720	−0.300962
$817$	9.66404	0.338102
$818$	−0.358531	−0.0125357
$819$	0	0
$820$	0	0
$821$	−21.3321	−0.744494	−0.372247	−	0.928134i	$-0.621413\pi$
$821$	−21.3321	−0.744494	−0.372247	+	0.928134i	$0.621413\pi$
$822$	−41.1665	−1.43585
$823$	19.9585	0.695708	0.347854	−	0.937549i	$-0.386910\pi$
$823$	19.9585	0.695708	0.347854	+	0.937549i	$0.386910\pi$
$824$	−3.00760	−0.104775
$825$	0	0
$826$	−13.3405	−0.464176
$827$	−28.9529	−1.00679	−0.503395	−	0.864056i	$-0.667916\pi$
$827$	−28.9529	−1.00679	−0.503395	+	0.864056i	$0.667916\pi$
$828$	−7.27299	−0.252754
$829$	27.1011	0.941261	0.470631	−	0.882330i	$-0.344026\pi$
$829$	27.1011	0.941261	0.470631	+	0.882330i	$0.344026\pi$
$830$	0	0
$831$	−7.80505	−0.270754
$832$	0	0
$833$	−18.6266	−0.645372
$834$	34.8822	1.20787
$835$	0	0
$836$	2.34624	0.0811464
$837$	44.2820	1.53061
$838$	−19.3184	−0.667344
$839$	33.7213	1.16419	0.582095	−	0.813121i	$-0.302233\pi$
$839$	33.7213	1.16419	0.582095	+	0.813121i	$0.302233\pi$
$840$	0	0
$841$	−21.5119	−0.741790
$842$	−0.220426	−0.00759637
$843$	46.7006	1.60846
$844$	0.404566	0.0139258
$845$	0	0
$846$	67.9053	2.33463
$847$	−36.4365	−1.25197
$848$	−3.81853	−0.131129
$849$	−76.1384	−2.61306
$850$	0	0
$851$	−11.6694	−0.400021
$852$	37.7262	1.29248
$853$	30.0853	1.03010	0.515050	−	0.857160i	$-0.327773\pi$
$853$	30.0853	1.03010	0.515050	+	0.857160i	$0.327773\pi$
$854$	56.2445	1.92465
$855$	0	0
$856$	−5.47595	−0.187164
$857$	−21.6341	−0.739006	−0.369503	−	0.929230i	$-0.620472\pi$
$857$	−21.6341	−0.739006	−0.369503	+	0.929230i	$0.620472\pi$
$858$	0	0
$859$	6.57516	0.224342	0.112171	−	0.993689i	$-0.464220\pi$
$859$	6.57516	0.224342	0.112171	+	0.993689i	$0.464220\pi$
$860$	0	0
$861$	−56.0386	−1.90979
$862$	19.9696	0.680168
$863$	−45.3471	−1.54363	−0.771817	−	0.635844i	$-0.780652\pi$
$863$	−45.3471	−1.54363	−0.771817	+	0.635844i	$0.780652\pi$
$864$	10.5385	0.358528
$865$	0	0
$866$	31.4155	1.06754
$867$	−28.1414	−0.955730
$868$	15.5265	0.527003
$869$	15.1817	0.515004
$870$	0	0
$871$	0	0
$872$	11.8872	0.402551
$873$	−56.2812	−1.90483
$874$	2.48578	0.0840828
$875$	0	0
$876$	3.78962	0.128040
$877$	47.7890	1.61372	0.806859	−	0.590743i	$-0.201165\pi$
$877$	47.7890	1.61372	0.806859	+	0.590743i	$0.201165\pi$
$878$	−25.6139	−0.864427
$879$	64.6861	2.18181
$880$	0	0
$881$	12.4378	0.419040	0.209520	−	0.977804i	$-0.432810\pi$
$881$	12.4378	0.419040	0.209520	+	0.977804i	$0.432810\pi$
$882$	42.7939	1.44095
$883$	16.6655	0.560840	0.280420	−	0.959877i	$-0.409526\pi$
$883$	16.6655	0.560840	0.280420	+	0.959877i	$0.409526\pi$
$884$	0	0
$885$	0	0
$886$	28.4024	0.954197
$887$	−39.8662	−1.33857	−0.669287	−	0.743004i	$-0.733400\pi$
$887$	−39.8662	−1.33857	−0.669287	+	0.743004i	$0.733400\pi$
$888$	31.6913	1.06349
$889$	27.3584	0.917572
$890$	0	0
$891$	−13.9504	−0.467356
$892$	0.832022	0.0278581
$893$	−23.2088	−0.776654
$894$	−21.5016	−0.719122
$895$	0	0
$896$	3.69510	0.123445
$897$	0	0
$898$	−22.4217	−0.748222
$899$	−11.4983	−0.383489
$900$	0	0
$901$	10.6896	0.356122
$902$	5.27082	0.175499
$903$	−49.8898	−1.66023
$904$	9.26298	0.308082
$905$	0	0
$906$	−0.597197	−0.0198405
$907$	−12.2772	−0.407658	−0.203829	−	0.979007i	$-0.565339\pi$
$907$	−12.2772	−0.407658	−0.203829	+	0.979007i	$0.565339\pi$
$908$	−7.77546	−0.258038
$909$	−16.6116	−0.550971
$910$	0	0
$911$	−47.4729	−1.57285	−0.786423	−	0.617688i	$-0.788070\pi$
$911$	−47.4729	−1.57285	−0.786423	+	0.617688i	$0.788070\pi$
$912$	−6.75081	−0.223542
$913$	8.73345	0.289035
$914$	23.5987	0.780574
$915$	0	0
$916$	7.73762	0.255658
$917$	−27.1896	−0.897880
$918$	−29.5016	−0.973698
$919$	12.0621	0.397892	0.198946	−	0.980010i	$-0.436248\pi$
$919$	12.0621	0.397892	0.198946	+	0.980010i	$0.436248\pi$
$920$	0	0
$921$	−27.3075	−0.899812
$922$	−32.5063	−1.07054
$923$	0	0
$924$	−12.1123	−0.398464
$925$	0	0
$926$	−7.70367	−0.253158
$927$	−19.3435	−0.635324
$928$	−2.73644	−0.0898280
$929$	−19.2582	−0.631840	−0.315920	−	0.948786i	$-0.602313\pi$
$929$	−19.2582	−0.631840	−0.315920	+	0.948786i	$0.602313\pi$
$930$	0	0
$931$	−14.6262	−0.479354
$932$	−6.86070	−0.224730
$933$	−88.1446	−2.88573
$934$	−20.9339	−0.684977
$935$	0	0
$936$	0	0
$937$	−17.1012	−0.558671	−0.279335	−	0.960194i	$-0.590114\pi$
$937$	−17.1012	−0.558671	−0.279335	+	0.960194i	$0.590114\pi$
$938$	14.7804	0.482597
$939$	24.0810	0.785853
$940$	0	0
$941$	−12.5077	−0.407741	−0.203870	−	0.978998i	$-0.565352\pi$
$941$	−12.5077	−0.407741	−0.203870	+	0.978998i	$0.565352\pi$
$942$	−51.8672	−1.68992
$943$	5.58430	0.181850
$944$	−3.61033	−0.117506
$945$	0	0
$946$	4.69248	0.152566
$947$	−60.2398	−1.95753	−0.978766	−	0.204983i	$-0.934286\pi$
$947$	−60.2398	−1.95753	−0.978766	+	0.204983i	$0.934286\pi$
$948$	−43.6821	−1.41873
$949$	0	0
$950$	0	0
$951$	−18.9713	−0.615187
$952$	−10.3441	−0.335253
$953$	−11.6460	−0.377251	−0.188625	−	0.982049i	$-0.560403\pi$
$953$	−11.6460	−0.377251	−0.188625	+	0.982049i	$0.560403\pi$
$954$	−24.5590	−0.795127
$955$	0	0
$956$	−22.3277	−0.722130
$957$	8.96985	0.289954
$958$	0.980499	0.0316785
$959$	−49.5312	−1.59945
$960$	0	0
$961$	−13.3439	−0.430449
$962$	0	0
$963$	−35.2188	−1.13491
$964$	−5.68717	−0.183172
$965$	0	0
$966$	−12.8326	−0.412883
$967$	2.95604	0.0950599	0.0475300	−	0.998870i	$-0.484865\pi$
$967$	2.95604	0.0950599	0.0475300	+	0.998870i	$0.484865\pi$
$968$	−9.86076	−0.316937
$969$	18.8982	0.607099
$970$	0	0
$971$	32.2421	1.03470	0.517350	−	0.855774i	$-0.326919\pi$
$971$	32.2421	1.03470	0.517350	+	0.855774i	$0.326919\pi$
$972$	8.52366	0.273397
$973$	41.9700	1.34550
$974$	38.6283	1.23773
$975$	0	0
$976$	15.2214	0.487224
$977$	37.2619	1.19211	0.596056	−	0.802943i	$-0.296734\pi$
$977$	37.2619	1.19211	0.596056	+	0.802943i	$0.296734\pi$
$978$	6.85358	0.219153
$979$	−9.39802	−0.300362
$980$	0	0
$981$	76.4530	2.44096
$982$	−17.7068	−0.565048
$983$	−31.3081	−0.998574	−0.499287	−	0.866437i	$-0.666405\pi$
$983$	−31.3081	−0.998574	−0.499287	+	0.866437i	$0.666405\pi$
$984$	−15.1657	−0.483464
$985$	0	0
$986$	7.66040	0.243957
$987$	119.814	3.81371
$988$	0	0
$989$	4.97156	0.158086
$990$	0	0
$991$	−8.16536	−0.259381	−0.129691	−	0.991555i	$-0.541398\pi$
$991$	−8.16536	−0.259381	−0.129691	+	0.991555i	$0.541398\pi$
$992$	4.20191	0.133411
$993$	81.8646	2.59789
$994$	45.3918	1.43974
$995$	0	0
$996$	−25.1286	−0.796232
$997$	47.5225	1.50505	0.752526	−	0.658563i	$-0.228835\pi$
$997$	47.5225	1.50505	0.752526	+	0.658563i	$0.228835\pi$
$998$	35.5402	1.12501
$999$	108.750	3.44070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cs.1.8		8
5.2	odd	4		1690.2.b.e.339.9		16
5.3	odd	4		1690.2.b.e.339.8		16
5.4	even	2		8450.2.a.cr.1.1		8
13.6	odd	12		650.2.m.e.101.1		16
13.11	odd	12		650.2.m.e.251.1		16
13.12	even	2		8450.2.a.cr.1.8		8
65.8	even	4		1690.2.c.f.1689.8		8
65.12	odd	4		1690.2.b.e.339.1		16
65.18	even	4		1690.2.c.e.1689.8		8
65.19	odd	12		650.2.m.e.101.8		16
65.24	odd	12		650.2.m.e.251.8		16
65.32	even	12		130.2.m.a.49.1	✓	8
65.37	even	12		130.2.m.b.69.4	yes	8
65.38	odd	4		1690.2.b.e.339.16		16
65.47	even	4		1690.2.c.e.1689.1		8
65.57	even	4		1690.2.c.f.1689.1		8
65.58	even	12		130.2.m.b.49.4	yes	8
65.63	even	12		130.2.m.a.69.1	yes	8
65.64	even	2	inner	8450.2.a.cs.1.1		8
195.32	odd	12		1170.2.bj.b.829.2		8
195.128	odd	12		1170.2.bj.b.199.2		8
195.167	odd	12		1170.2.bj.a.199.3		8
195.188	odd	12		1170.2.bj.a.829.3		8
260.63	odd	12		1040.2.df.c.849.4		8
260.123	odd	12		1040.2.df.a.49.1		8
260.167	odd	12		1040.2.df.a.849.1		8
260.227	odd	12		1040.2.df.c.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.m.a.49.1	✓	8	65.32	even	12
130.2.m.a.69.1	yes	8	65.63	even	12
130.2.m.b.49.4	yes	8	65.58	even	12
130.2.m.b.69.4	yes	8	65.37	even	12
650.2.m.e.101.1		16	13.6	odd	12
650.2.m.e.101.8		16	65.19	odd	12
650.2.m.e.251.1		16	13.11	odd	12
650.2.m.e.251.8		16	65.24	odd	12
1040.2.df.a.49.1		8	260.123	odd	12
1040.2.df.a.849.1		8	260.167	odd	12
1040.2.df.c.49.4		8	260.227	odd	12
1040.2.df.c.849.4		8	260.63	odd	12
1170.2.bj.a.199.3		8	195.167	odd	12
1170.2.bj.a.829.3		8	195.188	odd	12
1170.2.bj.b.199.2		8	195.128	odd	12
1170.2.bj.b.829.2		8	195.32	odd	12
1690.2.b.e.339.1		16	65.12	odd	4
1690.2.b.e.339.8		16	5.3	odd	4
1690.2.b.e.339.9		16	5.2	odd	4
1690.2.b.e.339.16		16	65.38	odd	4
1690.2.c.e.1689.1		8	65.47	even	4
1690.2.c.e.1689.8		8	65.18	even	4
1690.2.c.f.1689.1		8	65.57	even	4
1690.2.c.f.1689.8		8	65.8	even	4
8450.2.a.cr.1.1		8	5.4	even	2
8450.2.a.cr.1.8		8	13.12	even	2
8450.2.a.cs.1.1		8	65.64	even	2	inner
8450.2.a.cs.1.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.07108 q^{3} +1.00000 q^{4} +3.07108 q^{6} +3.69510 q^{7} +1.00000 q^{8} +6.43154 q^{9} -1.06735 q^{11} +3.07108 q^{12} +3.69510 q^{14} +1.00000 q^{16} -2.79940 q^{17} +6.43154 q^{18} -2.19819 q^{19} +11.3479 q^{21} -1.06735 q^{22} -1.13083 q^{23} +3.07108 q^{24} +10.5385 q^{27} +3.69510 q^{28} -2.73644 q^{29} +4.20191 q^{31} +1.00000 q^{32} -3.27793 q^{33} -2.79940 q^{34} +6.43154 q^{36} +10.3193 q^{37} -2.19819 q^{38} -4.93821 q^{41} +11.3479 q^{42} -4.39637 q^{43} -1.06735 q^{44} -1.13083 q^{46} +10.5582 q^{47} +3.07108 q^{48} +6.65376 q^{49} -8.59720 q^{51} -3.81853 q^{53} +10.5385 q^{54} +3.69510 q^{56} -6.75081 q^{57} -2.73644 q^{58} -3.61033 q^{59} +15.2214 q^{61} +4.20191 q^{62} +23.7652 q^{63} +1.00000 q^{64} -3.27793 q^{66} +4.00000 q^{67} -2.79940 q^{68} -3.47288 q^{69} +12.2843 q^{71} +6.43154 q^{72} +1.23397 q^{73} +10.3193 q^{74} -2.19819 q^{76} -3.94398 q^{77} -14.2237 q^{79} +13.0701 q^{81} -4.93821 q^{82} -8.18235 q^{83} +11.3479 q^{84} -4.39637 q^{86} -8.40383 q^{87} -1.06735 q^{88} +8.80497 q^{89} -1.13083 q^{92} +12.9044 q^{93} +10.5582 q^{94} +3.07108 q^{96} -8.75081 q^{97} +6.65376 q^{98} -6.86472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more