LMFDB - Embedded newform 9300.2.a.bf.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(1,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,7,0,0,0,4,0,7,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 15x^{5} + 49x^{4} + 13x^{3} - 69x^{2} - 35x - 1 $ x^7 - 3x^6 - 15x^5 + 49x^4 + 13x^3 - 69x^2 - 35x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.575791$ of defining polynomial
Character	$\chi$	$=$	9300.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.50165 q^{7} +1.00000 q^{9} +5.05534 q^{11} +2.57132 q^{13} +1.61312 q^{17} -4.52342 q^{19} +1.50165 q^{21} -1.39394 q^{23} +1.00000 q^{27} +5.33602 q^{29} -1.00000 q^{31} +5.05534 q^{33} +6.26484 q^{37} +2.57132 q^{39} +11.3596 q^{41} +3.19986 q^{43} +0.461540 q^{47} -4.74504 q^{49} +1.61312 q^{51} -1.97824 q^{53} -4.52342 q^{57} +0.593417 q^{59} -3.59136 q^{61} +1.50165 q^{63} -10.9852 q^{67} -1.39394 q^{69} +0.506127 q^{71} +1.89004 q^{73} +7.59136 q^{77} -8.33417 q^{79} +1.00000 q^{81} +1.12914 q^{83} +5.33602 q^{87} +6.08820 q^{89} +3.86122 q^{91} -1.00000 q^{93} +14.5355 q^{97} +5.05534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 10 q^{13} + 6 q^{17} + 10 q^{19} + 4 q^{21} + 7 q^{27} + 10 q^{29} - 7 q^{31} + 4 q^{33} + 16 q^{37} + 10 q^{39} + 2 q^{41} - 8 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 10 * q^13 + 6 * q^17 + 10 * q^19 + 4 * q^21 + 7 * q^27 + 10 * q^29 - 7 * q^31 + 4 * q^33 + 16 * q^37 + 10 * q^39 + 2 * q^41 - 8 * q^43 + 12 * q^47 + 9 * q^49 + 6 * q^51 + 16 * q^53 + 10 * q^57 + 14 * q^59 + 10 * q^61 + 4 * q^63 + 22 * q^67 - 2 * q^71 + 14 * q^73 + 18 * q^77 + 24 * q^79 + 7 * q^81 - 12 * q^83 + 10 * q^87 + 12 * q^89 + 8 * q^91 - 7 * q^93 + 24 * q^97 + 4 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.50165	0.567571	0.283786	−	0.958888i	$-0.408410\pi$
$7$	1.50165	0.567571	0.283786	+	0.958888i	$0.408410\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.05534	1.52424	0.762121	−	0.647434i	$-0.224158\pi$
$11$	5.05534	1.52424	0.762121	+	0.647434i	$0.224158\pi$
$12$	0	0
$13$	2.57132	0.713155	0.356577	−	0.934266i	$-0.383944\pi$
$13$	2.57132	0.713155	0.356577	+	0.934266i	$0.383944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.61312	0.391240	0.195620	−	0.980680i	$-0.437328\pi$
$17$	1.61312	0.391240	0.195620	+	0.980680i	$0.437328\pi$
$18$	0	0
$19$	−4.52342	−1.03774	−0.518872	−	0.854852i	$-0.673648\pi$
$19$	−4.52342	−1.03774	−0.518872	+	0.854852i	$0.673648\pi$
$20$	0	0
$21$	1.50165	0.327687
$22$	0	0
$23$	−1.39394	−0.290657	−0.145328	−	0.989383i	$-0.546424\pi$
$23$	−1.39394	−0.290657	−0.145328	+	0.989383i	$0.546424\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.33602	0.990874	0.495437	−	0.868644i	$-0.335008\pi$
$29$	5.33602	0.990874	0.495437	+	0.868644i	$0.335008\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	5.05534	0.880022
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.26484	1.02993	0.514967	−	0.857210i	$-0.327804\pi$
$37$	6.26484	1.02993	0.514967	+	0.857210i	$0.327804\pi$
$38$	0	0
$39$	2.57132	0.411740
$40$	0	0
$41$	11.3596	1.77407	0.887034	−	0.461704i	$-0.152762\pi$
$41$	11.3596	1.77407	0.887034	+	0.461704i	$0.152762\pi$
$42$	0	0
$43$	3.19986	0.487974	0.243987	−	0.969778i	$-0.421545\pi$
$43$	3.19986	0.487974	0.243987	+	0.969778i	$0.421545\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.461540	0.0673226	0.0336613	−	0.999433i	$-0.489283\pi$
$47$	0.461540	0.0673226	0.0336613	+	0.999433i	$0.489283\pi$
$48$	0	0
$49$	−4.74504	−0.677863
$50$	0	0
$51$	1.61312	0.225882
$52$	0	0
$53$	−1.97824	−0.271732	−0.135866	−	0.990727i	$-0.543382\pi$
$53$	−1.97824	−0.271732	−0.135866	+	0.990727i	$0.543382\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.52342	−0.599142
$58$	0	0
$59$	0.593417	0.0772563	0.0386282	−	0.999254i	$-0.487701\pi$
$59$	0.593417	0.0772563	0.0386282	+	0.999254i	$0.487701\pi$
$60$	0	0
$61$	−3.59136	−0.459827	−0.229914	−	0.973211i	$-0.573844\pi$
$61$	−3.59136	−0.459827	−0.229914	+	0.973211i	$0.573844\pi$
$62$	0	0
$63$	1.50165	0.189190
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.9852	−1.34205	−0.671026	−	0.741434i	$-0.734146\pi$
$67$	−10.9852	−1.34205	−0.671026	+	0.741434i	$0.734146\pi$
$68$	0	0
$69$	−1.39394	−0.167811
$70$	0	0
$71$	0.506127	0.0600663	0.0300331	−	0.999549i	$-0.490439\pi$
$71$	0.506127	0.0600663	0.0300331	+	0.999549i	$0.490439\pi$
$72$	0	0
$73$	1.89004	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$73$	1.89004	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.59136	0.865116
$78$	0	0
$79$	−8.33417	−0.937668	−0.468834	−	0.883286i	$-0.655326\pi$
$79$	−8.33417	−0.937668	−0.468834	+	0.883286i	$0.655326\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.12914	0.123939	0.0619695	−	0.998078i	$-0.480262\pi$
$83$	1.12914	0.123939	0.0619695	+	0.998078i	$0.480262\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.33602	0.572081
$88$	0	0
$89$	6.08820	0.645348	0.322674	−	0.946510i	$-0.395418\pi$
$89$	6.08820	0.645348	0.322674	+	0.946510i	$0.395418\pi$
$90$	0	0
$91$	3.86122	0.404766
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.5355	1.47586	0.737929	−	0.674879i	$-0.235804\pi$
$97$	14.5355	1.47586	0.737929	+	0.674879i	$0.235804\pi$
$98$	0	0
$99$	5.05534	0.508081
$100$	0	0
$101$	−17.2437	−1.71582	−0.857908	−	0.513803i	$-0.828236\pi$
$101$	−17.2437	−1.71582	−0.857908	+	0.513803i	$0.828236\pi$
$102$	0	0
$103$	14.4329	1.42212	0.711060	−	0.703132i	$-0.248216\pi$
$103$	14.4329	1.42212	0.711060	+	0.703132i	$0.248216\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.4104	−1.48978	−0.744892	−	0.667185i	$-0.767499\pi$
$107$	−15.4104	−1.48978	−0.744892	+	0.667185i	$0.767499\pi$
$108$	0	0
$109$	3.85871	0.369598	0.184799	−	0.982776i	$-0.440837\pi$
$109$	3.85871	0.369598	0.184799	+	0.982776i	$0.440837\pi$
$110$	0	0
$111$	6.26484	0.594633
$112$	0	0
$113$	10.2127	0.960734	0.480367	−	0.877067i	$-0.340503\pi$
$113$	10.2127	0.960734	0.480367	+	0.877067i	$0.340503\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.57132	0.237718
$118$	0	0
$119$	2.42235	0.222056
$120$	0	0
$121$	14.5565	1.32332
$122$	0	0
$123$	11.3596	1.02426
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.22060	0.551989	0.275994	−	0.961159i	$-0.410993\pi$
$127$	6.22060	0.551989	0.275994	+	0.961159i	$0.410993\pi$
$128$	0	0
$129$	3.19986	0.281732
$130$	0	0
$131$	−6.91702	−0.604343	−0.302171	−	0.953254i	$-0.597711\pi$
$131$	−6.91702	−0.604343	−0.302171	+	0.953254i	$0.597711\pi$
$132$	0	0
$133$	−6.79260	−0.588993
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.8735	−1.01442	−0.507211	−	0.861822i	$-0.669323\pi$
$137$	−11.8735	−1.01442	−0.507211	+	0.861822i	$0.669323\pi$
$138$	0	0
$139$	6.51340	0.552459	0.276230	−	0.961092i	$-0.410915\pi$
$139$	6.51340	0.552459	0.276230	+	0.961092i	$0.410915\pi$
$140$	0	0
$141$	0.461540	0.0388687
$142$	0	0
$143$	12.9989	1.08702
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.74504	−0.391364
$148$	0	0
$149$	7.72670	0.632996	0.316498	−	0.948593i	$-0.397493\pi$
$149$	7.72670	0.632996	0.316498	+	0.948593i	$0.397493\pi$
$150$	0	0
$151$	17.5567	1.42874	0.714370	−	0.699768i	$-0.246713\pi$
$151$	17.5567	1.42874	0.714370	+	0.699768i	$0.246713\pi$
$152$	0	0
$153$	1.61312	0.130413
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.07531	0.484863	0.242431	−	0.970169i	$-0.422055\pi$
$157$	6.07531	0.484863	0.242431	+	0.970169i	$0.422055\pi$
$158$	0	0
$159$	−1.97824	−0.156885
$160$	0	0
$161$	−2.09321	−0.164968
$162$	0	0
$163$	−0.960755	−0.0752521	−0.0376261	−	0.999292i	$-0.511980\pi$
$163$	−0.960755	−0.0752521	−0.0376261	+	0.999292i	$0.511980\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.7693	−0.833356	−0.416678	−	0.909054i	$-0.636806\pi$
$167$	−10.7693	−0.833356	−0.416678	+	0.909054i	$0.636806\pi$
$168$	0	0
$169$	−6.38834	−0.491410
$170$	0	0
$171$	−4.52342	−0.345915
$172$	0	0
$173$	1.78316	0.135571	0.0677856	−	0.997700i	$-0.478407\pi$
$173$	1.78316	0.135571	0.0677856	+	0.997700i	$0.478407\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.593417	0.0446040
$178$	0	0
$179$	0.206474	0.0154326	0.00771628	−	0.999970i	$-0.497544\pi$
$179$	0.206474	0.0154326	0.00771628	+	0.999970i	$0.497544\pi$
$180$	0	0
$181$	−6.23028	−0.463093	−0.231547	−	0.972824i	$-0.574379\pi$
$181$	−6.23028	−0.463093	−0.231547	+	0.972824i	$0.574379\pi$
$182$	0	0
$183$	−3.59136	−0.265481
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.15489	0.596344
$188$	0	0
$189$	1.50165	0.109229
$190$	0	0
$191$	5.00171	0.361911	0.180955	−	0.983491i	$-0.442081\pi$
$191$	5.00171	0.361911	0.180955	+	0.983491i	$0.442081\pi$
$192$	0	0
$193$	−24.9052	−1.79271	−0.896356	−	0.443335i	$-0.853795\pi$
$193$	−24.9052	−1.79271	−0.896356	+	0.443335i	$0.853795\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.5251	1.53360	0.766799	−	0.641887i	$-0.221848\pi$
$197$	21.5251	1.53360	0.766799	+	0.641887i	$0.221848\pi$
$198$	0	0
$199$	−5.36236	−0.380128	−0.190064	−	0.981772i	$-0.560870\pi$
$199$	−5.36236	−0.380128	−0.190064	+	0.981772i	$0.560870\pi$
$200$	0	0
$201$	−10.9852	−0.774834
$202$	0	0
$203$	8.01285	0.562392
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.39394	−0.0968855
$208$	0	0
$209$	−22.8674	−1.58177
$210$	0	0
$211$	−10.6244	−0.731415	−0.365708	−	0.930730i	$-0.619173\pi$
$211$	−10.6244	−0.731415	−0.365708	+	0.930730i	$0.619173\pi$
$212$	0	0
$213$	0.506127	0.0346793
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.50165	−0.101939
$218$	0	0
$219$	1.89004	0.127717
$220$	0	0
$221$	4.14785	0.279014
$222$	0	0
$223$	−8.50187	−0.569327	−0.284664	−	0.958627i	$-0.591882\pi$
$223$	−8.50187	−0.569327	−0.284664	+	0.958627i	$0.591882\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.59269	−0.570317	−0.285158	−	0.958480i	$-0.592046\pi$
$227$	−8.59269	−0.570317	−0.285158	+	0.958480i	$0.592046\pi$
$228$	0	0
$229$	−8.82248	−0.583006	−0.291503	−	0.956570i	$-0.594155\pi$
$229$	−8.82248	−0.583006	−0.291503	+	0.956570i	$0.594155\pi$
$230$	0	0
$231$	7.59136	0.499475
$232$	0	0
$233$	−15.5020	−1.01557	−0.507784	−	0.861485i	$-0.669535\pi$
$233$	−15.5020	−1.01557	−0.507784	+	0.861485i	$0.669535\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.33417	−0.541363
$238$	0	0
$239$	−14.9019	−0.963924	−0.481962	−	0.876192i	$-0.660076\pi$
$239$	−14.9019	−0.963924	−0.481962	+	0.876192i	$0.660076\pi$
$240$	0	0
$241$	6.31326	0.406673	0.203336	−	0.979109i	$-0.434821\pi$
$241$	6.31326	0.406673	0.203336	+	0.979109i	$0.434821\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.6311	−0.740072
$248$	0	0
$249$	1.12914	0.0715562
$250$	0	0
$251$	−5.65332	−0.356834	−0.178417	−	0.983955i	$-0.557098\pi$
$251$	−5.65332	−0.356834	−0.178417	+	0.983955i	$0.557098\pi$
$252$	0	0
$253$	−7.04684	−0.443031
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.69473	0.292849	0.146425	−	0.989222i	$-0.453223\pi$
$257$	4.69473	0.292849	0.146425	+	0.989222i	$0.453223\pi$
$258$	0	0
$259$	9.40761	0.584561
$260$	0	0
$261$	5.33602	0.330291
$262$	0	0
$263$	15.6902	0.967499	0.483750	−	0.875206i	$-0.339275\pi$
$263$	15.6902	0.967499	0.483750	+	0.875206i	$0.339275\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.08820	0.372592
$268$	0	0
$269$	17.9747	1.09593	0.547967	−	0.836500i	$-0.315402\pi$
$269$	17.9747	1.09593	0.547967	+	0.836500i	$0.315402\pi$
$270$	0	0
$271$	−7.15961	−0.434915	−0.217458	−	0.976070i	$-0.569776\pi$
$271$	−7.15961	−0.434915	−0.217458	+	0.976070i	$0.569776\pi$
$272$	0	0
$273$	3.86122	0.233692
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.8934	1.25536	0.627682	−	0.778470i	$-0.284004\pi$
$277$	20.8934	1.25536	0.627682	+	0.778470i	$0.284004\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−19.4131	−1.15809	−0.579043	−	0.815297i	$-0.696574\pi$
$281$	−19.4131	−1.15809	−0.579043	+	0.815297i	$0.696574\pi$
$282$	0	0
$283$	14.0000	0.832212	0.416106	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832212	0.416106	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.0581	1.00691
$288$	0	0
$289$	−14.3978	−0.846931
$290$	0	0
$291$	14.5355	0.852087
$292$	0	0
$293$	−20.9282	−1.22264	−0.611321	−	0.791383i	$-0.709361\pi$
$293$	−20.9282	−1.22264	−0.611321	+	0.791383i	$0.709361\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.05534	0.293341
$298$	0	0
$299$	−3.58426	−0.207283
$300$	0	0
$301$	4.80508	0.276960
$302$	0	0
$303$	−17.2437	−0.990627
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.8245	1.64510	0.822550	−	0.568693i	$-0.192551\pi$
$307$	28.8245	1.64510	0.822550	+	0.568693i	$0.192551\pi$
$308$	0	0
$309$	14.4329	0.821061
$310$	0	0
$311$	−23.0638	−1.30783	−0.653914	−	0.756569i	$-0.726874\pi$
$311$	−23.0638	−1.30783	−0.653914	+	0.756569i	$0.726874\pi$
$312$	0	0
$313$	−9.90417	−0.559817	−0.279908	−	0.960027i	$-0.590304\pi$
$313$	−9.90417	−0.559817	−0.279908	+	0.960027i	$0.590304\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.09406	−0.398442	−0.199221	−	0.979955i	$-0.563841\pi$
$317$	−7.09406	−0.398442	−0.199221	+	0.979955i	$0.563841\pi$
$318$	0	0
$319$	26.9754	1.51033
$320$	0	0
$321$	−15.4104	−0.860127
$322$	0	0
$323$	−7.29683	−0.406007
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.85871	0.213387
$328$	0	0
$329$	0.693073	0.0382103
$330$	0	0
$331$	23.9143	1.31445	0.657223	−	0.753696i	$-0.271731\pi$
$331$	23.9143	1.31445	0.657223	+	0.753696i	$0.271731\pi$
$332$	0	0
$333$	6.26484	0.343311
$334$	0	0
$335$	0	0
$336$	0	0
$337$	31.4193	1.71152	0.855760	−	0.517373i	$-0.173090\pi$
$337$	31.4193	1.71152	0.855760	+	0.517373i	$0.173090\pi$
$338$	0	0
$339$	10.2127	0.554680
$340$	0	0
$341$	−5.05534	−0.273762
$342$	0	0
$343$	−17.6370	−0.952307
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.8555	1.97851	0.989253	−	0.146211i	$-0.0467080\pi$
$347$	36.8555	1.97851	0.989253	+	0.146211i	$0.0467080\pi$
$348$	0	0
$349$	−1.71294	−0.0916917	−0.0458459	−	0.998949i	$-0.514598\pi$
$349$	−1.71294	−0.0916917	−0.0458459	+	0.998949i	$0.514598\pi$
$350$	0	0
$351$	2.57132	0.137247
$352$	0	0
$353$	14.9977	0.798247	0.399123	−	0.916897i	$-0.369315\pi$
$353$	14.9977	0.798247	0.399123	+	0.916897i	$0.369315\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.42235	0.128204
$358$	0	0
$359$	21.0429	1.11060	0.555301	−	0.831650i	$-0.312603\pi$
$359$	21.0429	1.11060	0.555301	+	0.831650i	$0.312603\pi$
$360$	0	0
$361$	1.46133	0.0769123
$362$	0	0
$363$	14.5565	0.764017
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.0302	1.46316	0.731582	−	0.681754i	$-0.238782\pi$
$367$	28.0302	1.46316	0.731582	+	0.681754i	$0.238782\pi$
$368$	0	0
$369$	11.3596	0.591356
$370$	0	0
$371$	−2.97063	−0.154227
$372$	0	0
$373$	24.8821	1.28835	0.644173	−	0.764880i	$-0.277202\pi$
$373$	24.8821	1.28835	0.644173	+	0.764880i	$0.277202\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.7206	0.706647
$378$	0	0
$379$	35.1571	1.80590	0.902951	−	0.429744i	$-0.141396\pi$
$379$	35.1571	1.80590	0.902951	+	0.429744i	$0.141396\pi$
$380$	0	0
$381$	6.22060	0.318691
$382$	0	0
$383$	−34.1796	−1.74649	−0.873247	−	0.487277i	$-0.837990\pi$
$383$	−34.1796	−1.74649	−0.873247	+	0.487277i	$0.837990\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.19986	0.162658
$388$	0	0
$389$	−2.01761	−0.102297	−0.0511485	−	0.998691i	$-0.516288\pi$
$389$	−2.01761	−0.102297	−0.0511485	+	0.998691i	$0.516288\pi$
$390$	0	0
$391$	−2.24860	−0.113716
$392$	0	0
$393$	−6.91702	−0.348917
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.19734	0.411413	0.205706	−	0.978614i	$-0.434051\pi$
$397$	8.19734	0.411413	0.205706	+	0.978614i	$0.434051\pi$
$398$	0	0
$399$	−6.79260	−0.340056
$400$	0	0
$401$	−35.3245	−1.76402	−0.882011	−	0.471229i	$-0.843811\pi$
$401$	−35.3245	−1.76402	−0.882011	+	0.471229i	$0.843811\pi$
$402$	0	0
$403$	−2.57132	−0.128086
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.6709	1.56987
$408$	0	0
$409$	−17.0512	−0.843125	−0.421563	−	0.906799i	$-0.638518\pi$
$409$	−17.0512	−0.843125	−0.421563	+	0.906799i	$0.638518\pi$
$410$	0	0
$411$	−11.8735	−0.585677
$412$	0	0
$413$	0.891106	0.0438485
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.51340	0.318962
$418$	0	0
$419$	−13.5831	−0.663579	−0.331790	−	0.943353i	$-0.607652\pi$
$419$	−13.5831	−0.663579	−0.331790	+	0.943353i	$0.607652\pi$
$420$	0	0
$421$	−21.8920	−1.06695	−0.533476	−	0.845815i	$-0.679115\pi$
$421$	−21.8920	−1.06695	−0.533476	+	0.845815i	$0.679115\pi$
$422$	0	0
$423$	0.461540	0.0224409
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.39298	−0.260985
$428$	0	0
$429$	12.9989	0.627592
$430$	0	0
$431$	−9.11877	−0.439236	−0.219618	−	0.975586i	$-0.570481\pi$
$431$	−9.11877	−0.439236	−0.219618	+	0.975586i	$0.570481\pi$
$432$	0	0
$433$	35.9549	1.72788	0.863942	−	0.503591i	$-0.167988\pi$
$433$	35.9549	1.72788	0.863942	+	0.503591i	$0.167988\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.30538	0.301627
$438$	0	0
$439$	−4.85665	−0.231795	−0.115898	−	0.993261i	$-0.536974\pi$
$439$	−4.85665	−0.231795	−0.115898	+	0.993261i	$0.536974\pi$
$440$	0	0
$441$	−4.74504	−0.225954
$442$	0	0
$443$	−5.58092	−0.265157	−0.132579	−	0.991172i	$-0.542326\pi$
$443$	−5.58092	−0.265157	−0.132579	+	0.991172i	$0.542326\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.72670	0.365460
$448$	0	0
$449$	25.7833	1.21679	0.608396	−	0.793634i	$-0.291813\pi$
$449$	25.7833	1.21679	0.608396	+	0.793634i	$0.291813\pi$
$450$	0	0
$451$	57.4266	2.70411
$452$	0	0
$453$	17.5567	0.824884
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.23474	0.291649	0.145825	−	0.989310i	$-0.453417\pi$
$457$	6.23474	0.291649	0.145825	+	0.989310i	$0.453417\pi$
$458$	0	0
$459$	1.61312	0.0752941
$460$	0	0
$461$	−16.6205	−0.774092	−0.387046	−	0.922060i	$-0.626505\pi$
$461$	−16.6205	−0.774092	−0.387046	+	0.922060i	$0.626505\pi$
$462$	0	0
$463$	7.36083	0.342087	0.171043	−	0.985263i	$-0.445286\pi$
$463$	7.36083	0.342087	0.171043	+	0.985263i	$0.445286\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.7877	−0.499194	−0.249597	−	0.968350i	$-0.580298\pi$
$467$	−10.7877	−0.499194	−0.249597	+	0.968350i	$0.580298\pi$
$468$	0	0
$469$	−16.4959	−0.761710
$470$	0	0
$471$	6.07531	0.279936
$472$	0	0
$473$	16.1764	0.743791
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.97824	−0.0905774
$478$	0	0
$479$	−5.57225	−0.254603	−0.127301	−	0.991864i	$-0.540632\pi$
$479$	−5.57225	−0.254603	−0.127301	+	0.991864i	$0.540632\pi$
$480$	0	0
$481$	16.1089	0.734502
$482$	0	0
$483$	−2.09321	−0.0952445
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.62003	0.0734104	0.0367052	−	0.999326i	$-0.488314\pi$
$487$	1.62003	0.0734104	0.0367052	+	0.999326i	$0.488314\pi$
$488$	0	0
$489$	−0.960755	−0.0434468
$490$	0	0
$491$	−6.80091	−0.306921	−0.153460	−	0.988155i	$-0.549042\pi$
$491$	−6.80091	−0.306921	−0.153460	+	0.988155i	$0.549042\pi$
$492$	0	0
$493$	8.60766	0.387669
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.760027	0.0340919
$498$	0	0
$499$	−23.6494	−1.05869	−0.529346	−	0.848406i	$-0.677563\pi$
$499$	−23.6494	−1.05869	−0.529346	+	0.848406i	$0.677563\pi$
$500$	0	0
$501$	−10.7693	−0.481138
$502$	0	0
$503$	34.2395	1.52667	0.763333	−	0.646005i	$-0.223562\pi$
$503$	34.2395	1.52667	0.763333	+	0.646005i	$0.223562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.38834	−0.283716
$508$	0	0
$509$	−1.41296	−0.0626285	−0.0313142	−	0.999510i	$-0.509969\pi$
$509$	−1.41296	−0.0626285	−0.0313142	+	0.999510i	$0.509969\pi$
$510$	0	0
$511$	2.83819	0.125554
$512$	0	0
$513$	−4.52342	−0.199714
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.33324	0.102616
$518$	0	0
$519$	1.78316	0.0782720
$520$	0	0
$521$	−12.3121	−0.539402	−0.269701	−	0.962944i	$-0.586925\pi$
$521$	−12.3121	−0.539402	−0.269701	+	0.962944i	$0.586925\pi$
$522$	0	0
$523$	−1.92693	−0.0842588	−0.0421294	−	0.999112i	$-0.513414\pi$
$523$	−1.92693	−0.0842588	−0.0421294	+	0.999112i	$0.513414\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.61312	−0.0702687
$528$	0	0
$529$	−21.0569	−0.915519
$530$	0	0
$531$	0.593417	0.0257521
$532$	0	0
$533$	29.2091	1.26519
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.206474	0.00891000
$538$	0	0
$539$	−23.9878	−1.03323
$540$	0	0
$541$	−1.31045	−0.0563408	−0.0281704	−	0.999603i	$-0.508968\pi$
$541$	−1.31045	−0.0563408	−0.0281704	+	0.999603i	$0.508968\pi$
$542$	0	0
$543$	−6.23028	−0.267367
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.38753	−0.315868	−0.157934	−	0.987450i	$-0.550483\pi$
$547$	−7.38753	−0.315868	−0.157934	+	0.987450i	$0.550483\pi$
$548$	0	0
$549$	−3.59136	−0.153276
$550$	0	0
$551$	−24.1371	−1.02827
$552$	0	0
$553$	−12.5150	−0.532193
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.9694	1.31222	0.656108	−	0.754667i	$-0.272202\pi$
$557$	30.9694	1.31222	0.656108	+	0.754667i	$0.272202\pi$
$558$	0	0
$559$	8.22786	0.348001
$560$	0	0
$561$	8.15489	0.344300
$562$	0	0
$563$	−8.06057	−0.339713	−0.169856	−	0.985469i	$-0.554330\pi$
$563$	−8.06057	−0.339713	−0.169856	+	0.985469i	$0.554330\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.50165	0.0630635
$568$	0	0
$569$	13.5493	0.568017	0.284008	−	0.958822i	$-0.408336\pi$
$569$	13.5493	0.568017	0.284008	+	0.958822i	$0.408336\pi$
$570$	0	0
$571$	−1.56539	−0.0655096	−0.0327548	−	0.999463i	$-0.510428\pi$
$571$	−1.56539	−0.0655096	−0.0327548	+	0.999463i	$0.510428\pi$
$572$	0	0
$573$	5.00171	0.208949
$574$	0	0
$575$	0	0
$576$	0	0
$577$	36.3272	1.51232	0.756161	−	0.654386i	$-0.227073\pi$
$577$	36.3272	1.51232	0.756161	+	0.654386i	$0.227073\pi$
$578$	0	0
$579$	−24.9052	−1.03502
$580$	0	0
$581$	1.69557	0.0703442
$582$	0	0
$583$	−10.0007	−0.414186
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.0931	1.48972	0.744861	−	0.667219i	$-0.232516\pi$
$587$	36.0931	1.48972	0.744861	+	0.667219i	$0.232516\pi$
$588$	0	0
$589$	4.52342	0.186384
$590$	0	0
$591$	21.5251	0.885424
$592$	0	0
$593$	−7.43742	−0.305418	−0.152709	−	0.988271i	$-0.548800\pi$
$593$	−7.43742	−0.305418	−0.152709	+	0.988271i	$0.548800\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.36236	−0.219467
$598$	0	0
$599$	−36.6070	−1.49572	−0.747860	−	0.663856i	$-0.768919\pi$
$599$	−36.6070	−1.49572	−0.747860	+	0.663856i	$0.768919\pi$
$600$	0	0
$601$	8.89587	0.362870	0.181435	−	0.983403i	$-0.441926\pi$
$601$	8.89587	0.362870	0.181435	+	0.983403i	$0.441926\pi$
$602$	0	0
$603$	−10.9852	−0.447350
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.9233	−0.443364	−0.221682	−	0.975119i	$-0.571155\pi$
$607$	−10.9233	−0.443364	−0.221682	+	0.975119i	$0.571155\pi$
$608$	0	0
$609$	8.01285	0.324697
$610$	0	0
$611$	1.18677	0.0480114
$612$	0	0
$613$	−36.5822	−1.47754	−0.738770	−	0.673958i	$-0.764593\pi$
$613$	−36.5822	−1.47754	−0.738770	+	0.673958i	$0.764593\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.8370	0.758351	0.379175	−	0.925325i	$-0.376208\pi$
$617$	18.8370	0.758351	0.379175	+	0.925325i	$0.376208\pi$
$618$	0	0
$619$	33.3099	1.33884	0.669420	−	0.742884i	$-0.266543\pi$
$619$	33.3099	1.33884	0.669420	+	0.742884i	$0.266543\pi$
$620$	0	0
$621$	−1.39394	−0.0559369
$622$	0	0
$623$	9.14235	0.366281
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−22.8674	−0.913237
$628$	0	0
$629$	10.1060	0.402951
$630$	0	0
$631$	35.8693	1.42793	0.713967	−	0.700179i	$-0.246897\pi$
$631$	35.8693	1.42793	0.713967	+	0.700179i	$0.246897\pi$
$632$	0	0
$633$	−10.6244	−0.422283
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.2010	−0.483421
$638$	0	0
$639$	0.506127	0.0200221
$640$	0	0
$641$	−31.1576	−1.23065	−0.615326	−	0.788273i	$-0.710976\pi$
$641$	−31.1576	−1.23065	−0.615326	+	0.788273i	$0.710976\pi$
$642$	0	0
$643$	−7.74588	−0.305468	−0.152734	−	0.988267i	$-0.548808\pi$
$643$	−7.74588	−0.305468	−0.152734	+	0.988267i	$0.548808\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.7517	0.579950	0.289975	−	0.957034i	$-0.406353\pi$
$647$	14.7517	0.579950	0.289975	+	0.957034i	$0.406353\pi$
$648$	0	0
$649$	2.99993	0.117757
$650$	0	0
$651$	−1.50165	−0.0588544
$652$	0	0
$653$	28.6804	1.12235	0.561176	−	0.827696i	$-0.310349\pi$
$653$	28.6804	1.12235	0.561176	+	0.827696i	$0.310349\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.89004	0.0737376
$658$	0	0
$659$	−43.9167	−1.71075	−0.855376	−	0.518008i	$-0.826674\pi$
$659$	−43.9167	−1.71075	−0.855376	+	0.518008i	$0.826674\pi$
$660$	0	0
$661$	−34.4640	−1.34049	−0.670247	−	0.742138i	$-0.733812\pi$
$661$	−34.4640	−1.34049	−0.670247	+	0.742138i	$0.733812\pi$
$662$	0	0
$663$	4.14785	0.161089
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.43809	−0.288004
$668$	0	0
$669$	−8.50187	−0.328701
$670$	0	0
$671$	−18.1556	−0.700888
$672$	0	0
$673$	39.9488	1.53991	0.769956	−	0.638097i	$-0.220278\pi$
$673$	39.9488	1.53991	0.769956	+	0.638097i	$0.220278\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.7045	1.87187	0.935933	−	0.352178i	$-0.114559\pi$
$677$	48.7045	1.87187	0.935933	+	0.352178i	$0.114559\pi$
$678$	0	0
$679$	21.8273	0.837654
$680$	0	0
$681$	−8.59269	−0.329272
$682$	0	0
$683$	−32.8639	−1.25750	−0.628752	−	0.777606i	$-0.716434\pi$
$683$	−32.8639	−1.25750	−0.628752	+	0.777606i	$0.716434\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.82248	−0.336599
$688$	0	0
$689$	−5.08668	−0.193787
$690$	0	0
$691$	−28.1285	−1.07006	−0.535029	−	0.844834i	$-0.679699\pi$
$691$	−28.1285	−1.07006	−0.535029	+	0.844834i	$0.679699\pi$
$692$	0	0
$693$	7.59136	0.288372
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.3244	0.694086
$698$	0	0
$699$	−15.5020	−0.586338
$700$	0	0
$701$	36.1314	1.36466	0.682331	−	0.731043i	$-0.260966\pi$
$701$	36.1314	1.36466	0.682331	+	0.731043i	$0.260966\pi$
$702$	0	0
$703$	−28.3385	−1.06881
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−25.8941	−0.973848
$708$	0	0
$709$	21.6999	0.814956	0.407478	−	0.913215i	$-0.366408\pi$
$709$	21.6999	0.814956	0.407478	+	0.913215i	$0.366408\pi$
$710$	0	0
$711$	−8.33417	−0.312556
$712$	0	0
$713$	1.39394	0.0522035
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.9019	−0.556522
$718$	0	0
$719$	8.54674	0.318740	0.159370	−	0.987219i	$-0.449054\pi$
$719$	8.54674	0.318740	0.159370	+	0.987219i	$0.449054\pi$
$720$	0	0
$721$	21.6732	0.807154
$722$	0	0
$723$	6.31326	0.234793
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.4390	−1.53689	−0.768443	−	0.639918i	$-0.778968\pi$
$727$	−41.4390	−1.53689	−0.768443	+	0.639918i	$0.778968\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.16177	0.190915
$732$	0	0
$733$	11.9883	0.442796	0.221398	−	0.975183i	$-0.428938\pi$
$733$	11.9883	0.442796	0.221398	+	0.975183i	$0.428938\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−55.5338	−2.04561
$738$	0	0
$739$	4.40364	0.161991	0.0809953	−	0.996714i	$-0.474190\pi$
$739$	4.40364	0.161991	0.0809953	+	0.996714i	$0.474190\pi$
$740$	0	0
$741$	−11.6311	−0.427281
$742$	0	0
$743$	−25.8842	−0.949600	−0.474800	−	0.880094i	$-0.657480\pi$
$743$	−25.8842	−0.949600	−0.474800	+	0.880094i	$0.657480\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.12914	0.0413130
$748$	0	0
$749$	−23.1411	−0.845558
$750$	0	0
$751$	36.6333	1.33677	0.668385	−	0.743816i	$-0.266986\pi$
$751$	36.6333	1.33677	0.668385	+	0.743816i	$0.266986\pi$
$752$	0	0
$753$	−5.65332	−0.206018
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.7059	0.498150	0.249075	−	0.968484i	$-0.419874\pi$
$757$	13.7059	0.498150	0.249075	+	0.968484i	$0.419874\pi$
$758$	0	0
$759$	−7.04684	−0.255784
$760$	0	0
$761$	20.8086	0.754310	0.377155	−	0.926150i	$-0.376902\pi$
$761$	20.8086	0.754310	0.377155	+	0.926150i	$0.376902\pi$
$762$	0	0
$763$	5.79445	0.209773
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.52586	0.0550957
$768$	0	0
$769$	−29.9958	−1.08168	−0.540838	−	0.841127i	$-0.681893\pi$
$769$	−29.9958	−1.08168	−0.540838	+	0.841127i	$0.681893\pi$
$770$	0	0
$771$	4.69473	0.169076
$772$	0	0
$773$	43.3059	1.55760	0.778802	−	0.627269i	$-0.215827\pi$
$773$	43.3059	1.55760	0.778802	+	0.627269i	$0.215827\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.40761	0.337496
$778$	0	0
$779$	−51.3842	−1.84103
$780$	0	0
$781$	2.55865	0.0915556
$782$	0	0
$783$	5.33602	0.190694
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−48.4959	−1.72869	−0.864346	−	0.502898i	$-0.832267\pi$
$787$	−48.4959	−1.72869	−0.864346	+	0.502898i	$0.832267\pi$
$788$	0	0
$789$	15.6902	0.558586
$790$	0	0
$791$	15.3360	0.545285
$792$	0	0
$793$	−9.23453	−0.327928
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.30223	0.329502	0.164751	−	0.986335i	$-0.447318\pi$
$797$	9.30223	0.329502	0.164751	+	0.986335i	$0.447318\pi$
$798$	0	0
$799$	0.744521	0.0263393
$800$	0	0
$801$	6.08820	0.215116
$802$	0	0
$803$	9.55482	0.337182
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.9747	0.632738
$808$	0	0
$809$	38.8841	1.36709	0.683546	−	0.729908i	$-0.260437\pi$
$809$	38.8841	1.36709	0.683546	+	0.729908i	$0.260437\pi$
$810$	0	0
$811$	33.6485	1.18156	0.590780	−	0.806833i	$-0.298820\pi$
$811$	33.6485	1.18156	0.590780	+	0.806833i	$0.298820\pi$
$812$	0	0
$813$	−7.15961	−0.251098
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.4743	−0.506392
$818$	0	0
$819$	3.86122	0.134922
$820$	0	0
$821$	−40.3217	−1.40724	−0.703619	−	0.710577i	$-0.748434\pi$
$821$	−40.3217	−1.40724	−0.703619	+	0.710577i	$0.748434\pi$
$822$	0	0
$823$	30.8884	1.07670	0.538351	−	0.842721i	$-0.319048\pi$
$823$	30.8884	1.07670	0.538351	+	0.842721i	$0.319048\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.5856	−1.44607	−0.723037	−	0.690809i	$-0.757255\pi$
$827$	−41.5856	−1.44607	−0.723037	+	0.690809i	$0.757255\pi$
$828$	0	0
$829$	−13.3957	−0.465251	−0.232625	−	0.972566i	$-0.574732\pi$
$829$	−13.3957	−0.465251	−0.232625	+	0.972566i	$0.574732\pi$
$830$	0	0
$831$	20.8934	0.724785
$832$	0	0
$833$	−7.65433	−0.265207
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	32.4085	1.11887	0.559433	−	0.828875i	$-0.311019\pi$
$839$	32.4085	1.11887	0.559433	+	0.828875i	$0.311019\pi$
$840$	0	0
$841$	−0.526880	−0.0181683
$842$	0	0
$843$	−19.4131	−0.668621
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.8588	0.751076
$848$	0	0
$849$	14.0000	0.480478
$850$	0	0
$851$	−8.73281	−0.299357
$852$	0	0
$853$	26.8787	0.920310	0.460155	−	0.887839i	$-0.347794\pi$
$853$	26.8787	0.920310	0.460155	+	0.887839i	$0.347794\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.5866	−1.31809	−0.659047	−	0.752102i	$-0.729040\pi$
$857$	−38.5866	−1.31809	−0.659047	+	0.752102i	$0.729040\pi$
$858$	0	0
$859$	39.7840	1.35741	0.678707	−	0.734409i	$-0.262541\pi$
$859$	39.7840	1.35741	0.678707	+	0.734409i	$0.262541\pi$
$860$	0	0
$861$	17.0581	0.581340
$862$	0	0
$863$	−49.8544	−1.69706	−0.848531	−	0.529145i	$-0.822513\pi$
$863$	−49.8544	−1.69706	−0.848531	+	0.529145i	$0.822513\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−14.3978	−0.488976
$868$	0	0
$869$	−42.1321	−1.42923
$870$	0	0
$871$	−28.2463	−0.957090
$872$	0	0
$873$	14.5355	0.491952
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.1950	0.648170	0.324085	−	0.946028i	$-0.394944\pi$
$877$	19.1950	0.648170	0.324085	+	0.946028i	$0.394944\pi$
$878$	0	0
$879$	−20.9282	−0.705892
$880$	0	0
$881$	−22.1421	−0.745988	−0.372994	−	0.927834i	$-0.621669\pi$
$881$	−22.1421	−0.745988	−0.372994	+	0.927834i	$0.621669\pi$
$882$	0	0
$883$	−3.43837	−0.115710	−0.0578552	−	0.998325i	$-0.518426\pi$
$883$	−3.43837	−0.115710	−0.0578552	+	0.998325i	$0.518426\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.54465	−0.0854411	−0.0427206	−	0.999087i	$-0.513603\pi$
$887$	−2.54465	−0.0854411	−0.0427206	+	0.999087i	$0.513603\pi$
$888$	0	0
$889$	9.34117	0.313293
$890$	0	0
$891$	5.05534	0.169360
$892$	0	0
$893$	−2.08774	−0.0698636
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.58426	−0.119675
$898$	0	0
$899$	−5.33602	−0.177966
$900$	0	0
$901$	−3.19115	−0.106312
$902$	0	0
$903$	4.80508	0.159903
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−47.4201	−1.57456	−0.787280	−	0.616596i	$-0.788511\pi$
$907$	−47.4201	−1.57456	−0.787280	+	0.616596i	$0.788511\pi$
$908$	0	0
$909$	−17.2437	−0.571939
$910$	0	0
$911$	15.4197	0.510879	0.255440	−	0.966825i	$-0.417780\pi$
$911$	15.4197	0.510879	0.255440	+	0.966825i	$0.417780\pi$
$912$	0	0
$913$	5.70818	0.188913
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.3870	−0.343007
$918$	0	0
$919$	−25.5422	−0.842559	−0.421280	−	0.906931i	$-0.638419\pi$
$919$	−25.5422	−0.842559	−0.421280	+	0.906931i	$0.638419\pi$
$920$	0	0
$921$	28.8245	0.949799
$922$	0	0
$923$	1.30141	0.0428365
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.4329	0.474040
$928$	0	0
$929$	33.7803	1.10830	0.554148	−	0.832418i	$-0.313044\pi$
$929$	33.7803	1.10830	0.554148	+	0.832418i	$0.313044\pi$
$930$	0	0
$931$	21.4638	0.703448
$932$	0	0
$933$	−23.0638	−0.755075
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21.3318	−0.696879	−0.348439	−	0.937331i	$-0.613288\pi$
$937$	−21.3318	−0.696879	−0.348439	+	0.937331i	$0.613288\pi$
$938$	0	0
$939$	−9.90417	−0.323210
$940$	0	0
$941$	−31.4967	−1.02676	−0.513381	−	0.858161i	$-0.671607\pi$
$941$	−31.4967	−1.02676	−0.513381	+	0.858161i	$0.671607\pi$
$942$	0	0
$943$	−15.8346	−0.515644
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.0808	−0.425068	−0.212534	−	0.977154i	$-0.568172\pi$
$947$	−13.0808	−0.425068	−0.212534	+	0.977154i	$0.568172\pi$
$948$	0	0
$949$	4.85990	0.157759
$950$	0	0
$951$	−7.09406	−0.230041
$952$	0	0
$953$	39.8336	1.29034	0.645169	−	0.764040i	$-0.276787\pi$
$953$	39.8336	1.29034	0.645169	+	0.764040i	$0.276787\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26.9754	0.871991
$958$	0	0
$959$	−17.8299	−0.575756
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−15.4104	−0.496595
$964$	0	0
$965$	0	0
$966$	0	0
$967$	15.7679	0.507063	0.253531	−	0.967327i	$-0.418408\pi$
$967$	15.7679	0.507063	0.253531	+	0.967327i	$0.418408\pi$
$968$	0	0
$969$	−7.29683	−0.234408
$970$	0	0
$971$	5.14966	0.165260	0.0826302	−	0.996580i	$-0.473668\pi$
$971$	5.14966	0.165260	0.0826302	+	0.996580i	$0.473668\pi$
$972$	0	0
$973$	9.78086	0.313560
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46.3077	−1.48152	−0.740758	−	0.671773i	$-0.765533\pi$
$977$	−46.3077	−1.48152	−0.740758	+	0.671773i	$0.765533\pi$
$978$	0	0
$979$	30.7779	0.983667
$980$	0	0
$981$	3.85871	0.123199
$982$	0	0
$983$	−6.50372	−0.207436	−0.103718	−	0.994607i	$-0.533074\pi$
$983$	−6.50372	−0.207436	−0.103718	+	0.994607i	$0.533074\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.693073	0.0220607
$988$	0	0
$989$	−4.46041	−0.141833
$990$	0	0
$991$	−38.6063	−1.22637	−0.613184	−	0.789940i	$-0.710112\pi$
$991$	−38.6063	−1.22637	−0.613184	+	0.789940i	$0.710112\pi$
$992$	0	0
$993$	23.9143	0.758896
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.1908	−0.829470	−0.414735	−	0.909942i	$-0.636126\pi$
$997$	−26.1908	−0.829470	−0.414735	+	0.909942i	$0.636126\pi$
$998$	0	0
$999$	6.26484	0.198211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.bf.1.5		7
5.2	odd	4		1860.2.g.a.1489.6	✓	14
5.3	odd	4		1860.2.g.a.1489.13	yes	14
5.4	even	2		9300.2.a.bc.1.3		7
15.2	even	4		5580.2.g.d.3349.4		14
15.8	even	4		5580.2.g.d.3349.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.g.a.1489.6	✓	14	5.2	odd	4
1860.2.g.a.1489.13	yes	14	5.3	odd	4
5580.2.g.d.3349.3		14	15.8	even	4
5580.2.g.d.3349.4		14	15.2	even	4
9300.2.a.bc.1.3		7	5.4	even	2
9300.2.a.bf.1.5		7	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:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.50165 q^{7} +1.00000 q^{9} +5.05534 q^{11} +2.57132 q^{13} +1.61312 q^{17} -4.52342 q^{19} +1.50165 q^{21} -1.39394 q^{23} +1.00000 q^{27} +5.33602 q^{29} -1.00000 q^{31} +5.05534 q^{33} +6.26484 q^{37} +2.57132 q^{39} +11.3596 q^{41} +3.19986 q^{43} +0.461540 q^{47} -4.74504 q^{49} +1.61312 q^{51} -1.97824 q^{53} -4.52342 q^{57} +0.593417 q^{59} -3.59136 q^{61} +1.50165 q^{63} -10.9852 q^{67} -1.39394 q^{69} +0.506127 q^{71} +1.89004 q^{73} +7.59136 q^{77} -8.33417 q^{79} +1.00000 q^{81} +1.12914 q^{83} +5.33602 q^{87} +6.08820 q^{89} +3.86122 q^{91} -1.00000 q^{93} +14.5355 q^{97} +5.05534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 10 q^{13} + 6 q^{17} + 10 q^{19} + 4 q^{21} + 7 q^{27} + 10 q^{29} - 7 q^{31} + 4 q^{33} + 16 q^{37} + 10 q^{39} + 2 q^{41} - 8 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 10 * q^13 + 6 * q^17 + 10 * q^19 + 4 * q^21 + 7 * q^27 + 10 * q^29 - 7 * q^31 + 4 * q^33 + 16 * q^37 + 10 * q^39 + 2 * q^41 - 8 * q^43 + 12 * q^47 + 9 * q^49 + 6 * q^51 + 16 * q^53 + 10 * q^57 + 14 * q^59 + 10 * q^61 + 4 * q^63 + 22 * q^67 - 2 * q^71 + 14 * q^73 + 18 * q^77 + 24 * q^79 + 7 * q^81 - 12 * q^83 + 10 * q^87 + 12 * q^89 + 8 * q^91 - 7 * q^93 + 24 * q^97 + 4 * q^99

Introduction

L-functions

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