LMFDB - Embedded newform 6900.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$55.0967773947$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -0.539189 q^{11} -2.24846 q^{13} +3.24846 q^{17} +2.24846 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{27} -8.51026 q^{29} -6.70928 q^{31} +0.539189 q^{33} +7.04945 q^{37} +2.24846 q^{39} +5.14116 q^{41} -4.97107 q^{43} +8.87936 q^{47} -6.00000 q^{49} -3.24846 q^{51} -4.80098 q^{53} -2.24846 q^{57} -0.170086 q^{59} -2.67316 q^{61} +1.00000 q^{63} +8.31124 q^{67} +1.00000 q^{69} -11.2979 q^{71} -8.09171 q^{73} -0.539189 q^{77} +5.57531 q^{79} +1.00000 q^{81} +0.411363 q^{83} +8.51026 q^{87} -6.68035 q^{89} -2.24846 q^{91} +6.70928 q^{93} -14.8865 q^{97} -0.539189 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{13} + q^{17} - 2 q^{19} - 3 q^{21} - 3 q^{23} - 3 q^{27} - 9 q^{29} - 13 q^{31} + 3 q^{37} - 2 q^{39} - 5 q^{41} + 14 q^{47} - 18 q^{49} - q^{51} - 5 q^{53} + 2 q^{57} + 5 q^{59} - 20 q^{61} + 3 q^{63} - q^{67} + 3 q^{69} - 7 q^{71} - 22 q^{73} - 4 q^{79} + 3 q^{81} + 21 q^{83} + 9 q^{87} + 2 q^{89} + 2 q^{91} + 13 q^{93} + 2 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^13 + q^17 - 2 * q^19 - 3 * q^21 - 3 * q^23 - 3 * q^27 - 9 * q^29 - 13 * q^31 + 3 * q^37 - 2 * q^39 - 5 * q^41 + 14 * q^47 - 18 * q^49 - q^51 - 5 * q^53 + 2 * q^57 + 5 * q^59 - 20 * q^61 + 3 * q^63 - q^67 + 3 * q^69 - 7 * q^71 - 22 * q^73 - 4 * q^79 + 3 * q^81 + 21 * q^83 + 9 * q^87 + 2 * q^89 + 2 * q^91 + 13 * q^93 + 2 * q^97

Display 100 coefficients 10 coefficients

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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.539189	−0.162572	−0.0812858	−	0.996691i	$-0.525903\pi$
$11$	−0.539189	−0.162572	−0.0812858	+	0.996691i	$0.525903\pi$
$12$	0	0
$13$	−2.24846	−0.623612	−0.311806	−	0.950146i	$-0.600934\pi$
$13$	−2.24846	−0.623612	−0.311806	+	0.950146i	$0.600934\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.24846	0.787868	0.393934	−	0.919139i	$-0.371114\pi$
$17$	3.24846	0.787868	0.393934	+	0.919139i	$0.371114\pi$
$18$	0	0
$19$	2.24846	0.515833	0.257917	−	0.966167i	$-0.416964\pi$
$19$	2.24846	0.515833	0.257917	+	0.966167i	$0.416964\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.51026	−1.58032	−0.790158	−	0.612904i	$-0.790001\pi$
$29$	−8.51026	−1.58032	−0.790158	+	0.612904i	$0.790001\pi$
$30$	0	0
$31$	−6.70928	−1.20502	−0.602511	−	0.798111i	$-0.705833\pi$
$31$	−6.70928	−1.20502	−0.602511	+	0.798111i	$0.705833\pi$
$32$	0	0
$33$	0.539189	0.0938607
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.04945	1.15892	0.579461	−	0.815000i	$-0.303263\pi$
$37$	7.04945	1.15892	0.579461	+	0.815000i	$0.303263\pi$
$38$	0	0
$39$	2.24846	0.360042
$40$	0	0
$41$	5.14116	0.802914	0.401457	−	0.915878i	$-0.368504\pi$
$41$	5.14116	0.802914	0.401457	+	0.915878i	$0.368504\pi$
$42$	0	0
$43$	−4.97107	−0.758081	−0.379041	−	0.925380i	$-0.623746\pi$
$43$	−4.97107	−0.758081	−0.379041	+	0.925380i	$0.623746\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.87936	1.29519	0.647594	−	0.761986i	$-0.275775\pi$
$47$	8.87936	1.29519	0.647594	+	0.761986i	$0.275775\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.24846	−0.454876
$52$	0	0
$53$	−4.80098	−0.659466	−0.329733	−	0.944074i	$-0.606959\pi$
$53$	−4.80098	−0.659466	−0.329733	+	0.944074i	$0.606959\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.24846	−0.297816
$58$	0	0
$59$	−0.170086	−0.0221434	−0.0110717	−	0.999939i	$-0.503524\pi$
$59$	−0.170086	−0.0221434	−0.0110717	+	0.999939i	$0.503524\pi$
$60$	0	0
$61$	−2.67316	−0.342263	−0.171131	−	0.985248i	$-0.554742\pi$
$61$	−2.67316	−0.342263	−0.171131	+	0.985248i	$0.554742\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.31124	1.01538	0.507690	−	0.861540i	$-0.330500\pi$
$67$	8.31124	1.01538	0.507690	+	0.861540i	$0.330500\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−11.2979	−1.34082	−0.670408	−	0.741993i	$-0.733881\pi$
$71$	−11.2979	−1.34082	−0.670408	+	0.741993i	$0.733881\pi$
$72$	0	0
$73$	−8.09171	−0.947063	−0.473531	−	0.880777i	$-0.657021\pi$
$73$	−8.09171	−0.947063	−0.473531	+	0.880777i	$0.657021\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.539189	−0.0614463
$78$	0	0
$79$	5.57531	0.627271	0.313635	−	0.949543i	$-0.398453\pi$
$79$	5.57531	0.627271	0.313635	+	0.949543i	$0.398453\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.411363	0.0451529	0.0225765	−	0.999745i	$-0.492813\pi$
$83$	0.411363	0.0451529	0.0225765	+	0.999745i	$0.492813\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.51026	0.912396
$88$	0	0
$89$	−6.68035	−0.708115	−0.354058	−	0.935224i	$-0.615198\pi$
$89$	−6.68035	−0.708115	−0.354058	+	0.935224i	$0.615198\pi$
$90$	0	0
$91$	−2.24846	−0.235703
$92$	0	0
$93$	6.70928	0.695719
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.8865	−1.51150	−0.755750	−	0.654860i	$-0.772728\pi$
$97$	−14.8865	−1.51150	−0.755750	+	0.654860i	$0.772728\pi$
$98$	0	0
$99$	−0.539189	−0.0541905
$100$	0	0
$101$	−2.12064	−0.211011	−0.105506	−	0.994419i	$-0.533646\pi$
$101$	−2.12064	−0.211011	−0.105506	+	0.994419i	$0.533646\pi$
$102$	0	0
$103$	−0.290725	−0.0286460	−0.0143230	−	0.999897i	$-0.504559\pi$
$103$	−0.290725	−0.0286460	−0.0143230	+	0.999897i	$0.504559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.3763	1.19646	0.598231	−	0.801324i	$-0.295871\pi$
$107$	12.3763	1.19646	0.598231	+	0.801324i	$0.295871\pi$
$108$	0	0
$109$	12.1906	1.16765	0.583824	−	0.811880i	$-0.301556\pi$
$109$	12.1906	1.16765	0.583824	+	0.811880i	$0.301556\pi$
$110$	0	0
$111$	−7.04945	−0.669104
$112$	0	0
$113$	1.72261	0.162049	0.0810246	−	0.996712i	$-0.474181\pi$
$113$	1.72261	0.162049	0.0810246	+	0.996712i	$0.474181\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.24846	−0.207871
$118$	0	0
$119$	3.24846	0.297786
$120$	0	0
$121$	−10.7093	−0.973570
$122$	0	0
$123$	−5.14116	−0.463563
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.1639	−1.07938	−0.539688	−	0.841865i	$-0.681458\pi$
$127$	−12.1639	−1.07938	−0.539688	+	0.841865i	$0.681458\pi$
$128$	0	0
$129$	4.97107	0.437678
$130$	0	0
$131$	7.78539	0.680212	0.340106	−	0.940387i	$-0.389537\pi$
$131$	7.78539	0.680212	0.340106	+	0.940387i	$0.389537\pi$
$132$	0	0
$133$	2.24846	0.194967
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.86603	0.842912	0.421456	−	0.906849i	$-0.361519\pi$
$137$	9.86603	0.842912	0.421456	+	0.906849i	$0.361519\pi$
$138$	0	0
$139$	4.07838	0.345923	0.172962	−	0.984929i	$-0.444666\pi$
$139$	4.07838	0.345923	0.172962	+	0.984929i	$0.444666\pi$
$140$	0	0
$141$	−8.87936	−0.747777
$142$	0	0
$143$	1.21235	0.101382
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−8.53919	−0.699558	−0.349779	−	0.936832i	$-0.613743\pi$
$149$	−8.53919	−0.699558	−0.349779	+	0.936832i	$0.613743\pi$
$150$	0	0
$151$	−2.63090	−0.214099	−0.107050	−	0.994254i	$-0.534140\pi$
$151$	−2.63090	−0.214099	−0.107050	+	0.994254i	$0.534140\pi$
$152$	0	0
$153$	3.24846	0.262623
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.7587	1.33749	0.668746	−	0.743491i	$-0.266831\pi$
$157$	16.7587	1.33749	0.668746	+	0.743491i	$0.266831\pi$
$158$	0	0
$159$	4.80098	0.380743
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−5.26180	−0.412136	−0.206068	−	0.978538i	$-0.566067\pi$
$163$	−5.26180	−0.412136	−0.206068	+	0.978538i	$0.566067\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.4752	−1.50704	−0.753518	−	0.657428i	$-0.771644\pi$
$167$	−19.4752	−1.50704	−0.753518	+	0.657428i	$0.771644\pi$
$168$	0	0
$169$	−7.94441	−0.611108
$170$	0	0
$171$	2.24846	0.171944
$172$	0	0
$173$	20.4391	1.55395	0.776977	−	0.629529i	$-0.216752\pi$
$173$	20.4391	1.55395	0.776977	+	0.629529i	$0.216752\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.170086	0.0127845
$178$	0	0
$179$	2.78765	0.208359	0.104179	−	0.994559i	$-0.466778\pi$
$179$	2.78765	0.208359	0.104179	+	0.994559i	$0.466778\pi$
$180$	0	0
$181$	−15.9155	−1.18299	−0.591494	−	0.806309i	$-0.701462\pi$
$181$	−15.9155	−1.18299	−0.591494	+	0.806309i	$0.701462\pi$
$182$	0	0
$183$	2.67316	0.197606
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.75154	−0.128085
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−1.12064	−0.0810865	−0.0405433	−	0.999178i	$-0.512909\pi$
$191$	−1.12064	−0.0810865	−0.0405433	+	0.999178i	$0.512909\pi$
$192$	0	0
$193$	12.8638	0.925954	0.462977	−	0.886370i	$-0.346781\pi$
$193$	12.8638	0.925954	0.462977	+	0.886370i	$0.346781\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7854	−0.839674	−0.419837	−	0.907600i	$-0.637913\pi$
$197$	−11.7854	−0.839674	−0.419837	+	0.907600i	$0.637913\pi$
$198$	0	0
$199$	−25.1194	−1.78067	−0.890334	−	0.455308i	$-0.849529\pi$
$199$	−25.1194	−1.78067	−0.890334	+	0.455308i	$0.849529\pi$
$200$	0	0
$201$	−8.31124	−0.586230
$202$	0	0
$203$	−8.51026	−0.597303
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−1.21235	−0.0838598
$210$	0	0
$211$	4.51745	0.310994	0.155497	−	0.987836i	$-0.450302\pi$
$211$	4.51745	0.310994	0.155497	+	0.987836i	$0.450302\pi$
$212$	0	0
$213$	11.2979	0.774120
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.70928	−0.455455
$218$	0	0
$219$	8.09171	0.546787
$220$	0	0
$221$	−7.30406	−0.491324
$222$	0	0
$223$	−3.31965	−0.222300	−0.111150	−	0.993804i	$-0.535453\pi$
$223$	−3.31965	−0.222300	−0.111150	+	0.993804i	$0.535453\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.5174	1.29542	0.647709	−	0.761888i	$-0.275727\pi$
$227$	19.5174	1.29542	0.647709	+	0.761888i	$0.275727\pi$
$228$	0	0
$229$	−9.39189	−0.620633	−0.310317	−	0.950633i	$-0.600435\pi$
$229$	−9.39189	−0.620633	−0.310317	+	0.950633i	$0.600435\pi$
$230$	0	0
$231$	0.539189	0.0354760
$232$	0	0
$233$	−15.1506	−0.992550	−0.496275	−	0.868165i	$-0.665299\pi$
$233$	−15.1506	−0.992550	−0.496275	+	0.868165i	$0.665299\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.57531	−0.362155
$238$	0	0
$239$	−27.5536	−1.78229	−0.891146	−	0.453717i	$-0.850098\pi$
$239$	−27.5536	−1.78229	−0.891146	+	0.453717i	$0.850098\pi$
$240$	0	0
$241$	−18.6153	−1.19912	−0.599558	−	0.800331i	$-0.704657\pi$
$241$	−18.6153	−1.19912	−0.599558	+	0.800331i	$0.704657\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.05559	−0.321680
$248$	0	0
$249$	−0.411363	−0.0260691
$250$	0	0
$251$	−4.18342	−0.264055	−0.132027	−	0.991246i	$-0.542149\pi$
$251$	−4.18342	−0.264055	−0.132027	+	0.991246i	$0.542149\pi$
$252$	0	0
$253$	0.539189	0.0338985
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.9060	−0.805056	−0.402528	−	0.915408i	$-0.631868\pi$
$257$	−12.9060	−0.805056	−0.402528	+	0.915408i	$0.631868\pi$
$258$	0	0
$259$	7.04945	0.438031
$260$	0	0
$261$	−8.51026	−0.526772
$262$	0	0
$263$	4.70209	0.289943	0.144972	−	0.989436i	$-0.453691\pi$
$263$	4.70209	0.289943	0.144972	+	0.989436i	$0.453691\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.68035	0.408831
$268$	0	0
$269$	−30.5246	−1.86112	−0.930560	−	0.366140i	$-0.880679\pi$
$269$	−30.5246	−1.86112	−0.930560	+	0.366140i	$0.880679\pi$
$270$	0	0
$271$	−2.65983	−0.161573	−0.0807865	−	0.996731i	$-0.525743\pi$
$271$	−2.65983	−0.161573	−0.0807865	+	0.996731i	$0.525743\pi$
$272$	0	0
$273$	2.24846	0.136083
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.581449	0.0349359	0.0174680	−	0.999847i	$-0.494439\pi$
$277$	0.581449	0.0349359	0.0174680	+	0.999847i	$0.494439\pi$
$278$	0	0
$279$	−6.70928	−0.401674
$280$	0	0
$281$	29.3907	1.75330	0.876650	−	0.481129i	$-0.159773\pi$
$281$	29.3907	1.75330	0.876650	+	0.481129i	$0.159773\pi$
$282$	0	0
$283$	2.23513	0.132865	0.0664324	−	0.997791i	$-0.478838\pi$
$283$	2.23513	0.132865	0.0664324	+	0.997791i	$0.478838\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.14116	0.303473
$288$	0	0
$289$	−6.44748	−0.379264
$290$	0	0
$291$	14.8865	0.872665
$292$	0	0
$293$	6.01333	0.351303	0.175651	−	0.984452i	$-0.443797\pi$
$293$	6.01333	0.351303	0.175651	+	0.984452i	$0.443797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.539189	0.0312869
$298$	0	0
$299$	2.24846	0.130032
$300$	0	0
$301$	−4.97107	−0.286528
$302$	0	0
$303$	2.12064	0.121827
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.516403	−0.0294726	−0.0147363	−	0.999891i	$-0.504691\pi$
$307$	−0.516403	−0.0294726	−0.0147363	+	0.999891i	$0.504691\pi$
$308$	0	0
$309$	0.290725	0.0165387
$310$	0	0
$311$	−11.5031	−0.652279	−0.326140	−	0.945322i	$-0.605748\pi$
$311$	−11.5031	−0.652279	−0.326140	+	0.945322i	$0.605748\pi$
$312$	0	0
$313$	3.70313	0.209313	0.104657	−	0.994508i	$-0.466626\pi$
$313$	3.70313	0.209313	0.104657	+	0.994508i	$0.466626\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.9204	−1.06268	−0.531338	−	0.847160i	$-0.678310\pi$
$317$	−18.9204	−1.06268	−0.531338	+	0.847160i	$0.678310\pi$
$318$	0	0
$319$	4.58864	0.256914
$320$	0	0
$321$	−12.3763	−0.690777
$322$	0	0
$323$	7.30406	0.406409
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.1906	−0.674142
$328$	0	0
$329$	8.87936	0.489535
$330$	0	0
$331$	1.29072	0.0709446	0.0354723	−	0.999371i	$-0.488706\pi$
$331$	1.29072	0.0709446	0.0354723	+	0.999371i	$0.488706\pi$
$332$	0	0
$333$	7.04945	0.386307
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.5236	−0.682203	−0.341102	−	0.940026i	$-0.610800\pi$
$337$	−12.5236	−0.682203	−0.341102	+	0.940026i	$0.610800\pi$
$338$	0	0
$339$	−1.72261	−0.0935591
$340$	0	0
$341$	3.61757	0.195902
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.44521	0.292314	0.146157	−	0.989261i	$-0.453309\pi$
$347$	5.44521	0.292314	0.146157	+	0.989261i	$0.453309\pi$
$348$	0	0
$349$	−5.39803	−0.288950	−0.144475	−	0.989508i	$-0.546149\pi$
$349$	−5.39803	−0.288950	−0.144475	+	0.989508i	$0.546149\pi$
$350$	0	0
$351$	2.24846	0.120014
$352$	0	0
$353$	14.0566	0.748159	0.374080	−	0.927397i	$-0.377959\pi$
$353$	14.0566	0.748159	0.374080	+	0.927397i	$0.377959\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.24846	−0.171927
$358$	0	0
$359$	27.9143	1.47326	0.736629	−	0.676297i	$-0.236416\pi$
$359$	27.9143	1.47326	0.736629	+	0.676297i	$0.236416\pi$
$360$	0	0
$361$	−13.9444	−0.733916
$362$	0	0
$363$	10.7093	0.562091
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.6248	−1.07660	−0.538302	−	0.842752i	$-0.680934\pi$
$367$	−20.6248	−1.07660	−0.538302	+	0.842752i	$0.680934\pi$
$368$	0	0
$369$	5.14116	0.267638
$370$	0	0
$371$	−4.80098	−0.249255
$372$	0	0
$373$	1.22672	0.0635173	0.0317586	−	0.999496i	$-0.489889\pi$
$373$	1.22672	0.0635173	0.0317586	+	0.999496i	$0.489889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.1350	0.985503
$378$	0	0
$379$	21.3607	1.09723	0.548613	−	0.836077i	$-0.315156\pi$
$379$	21.3607	1.09723	0.548613	+	0.836077i	$0.315156\pi$
$380$	0	0
$381$	12.1639	0.623178
$382$	0	0
$383$	31.9698	1.63358	0.816791	−	0.576933i	$-0.195751\pi$
$383$	31.9698	1.63358	0.816791	+	0.576933i	$0.195751\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.97107	−0.252694
$388$	0	0
$389$	18.3545	0.930613	0.465306	−	0.885150i	$-0.345944\pi$
$389$	18.3545	0.930613	0.465306	+	0.885150i	$0.345944\pi$
$390$	0	0
$391$	−3.24846	−0.164282
$392$	0	0
$393$	−7.78539	−0.392721
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.5753	−0.982456	−0.491228	−	0.871031i	$-0.663452\pi$
$397$	−19.5753	−0.982456	−0.491228	+	0.871031i	$0.663452\pi$
$398$	0	0
$399$	−2.24846	−0.112564
$400$	0	0
$401$	−9.50307	−0.474561	−0.237280	−	0.971441i	$-0.576256\pi$
$401$	−9.50307	−0.474561	−0.237280	+	0.971441i	$0.576256\pi$
$402$	0	0
$403$	15.0856	0.751466
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.80098	−0.188408
$408$	0	0
$409$	−29.8059	−1.47381	−0.736904	−	0.675998i	$-0.763713\pi$
$409$	−29.8059	−1.47381	−0.736904	+	0.675998i	$0.763713\pi$
$410$	0	0
$411$	−9.86603	−0.486655
$412$	0	0
$413$	−0.170086	−0.00836941
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.07838	−0.199719
$418$	0	0
$419$	−2.39681	−0.117092	−0.0585459	−	0.998285i	$-0.518646\pi$
$419$	−2.39681	−0.117092	−0.0585459	+	0.998285i	$0.518646\pi$
$420$	0	0
$421$	−29.2111	−1.42366	−0.711832	−	0.702350i	$-0.752134\pi$
$421$	−29.2111	−1.42366	−0.711832	+	0.702350i	$0.752134\pi$
$422$	0	0
$423$	8.87936	0.431729
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.67316	−0.129363
$428$	0	0
$429$	−1.21235	−0.0585327
$430$	0	0
$431$	−12.5236	−0.603240	−0.301620	−	0.953428i	$-0.597527\pi$
$431$	−12.5236	−0.603240	−0.301620	+	0.953428i	$0.597527\pi$
$432$	0	0
$433$	−28.0928	−1.35005	−0.675026	−	0.737794i	$-0.735868\pi$
$433$	−28.0928	−1.35005	−0.675026	+	0.737794i	$0.735868\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.24846	−0.107559
$438$	0	0
$439$	18.5997	0.887715	0.443858	−	0.896097i	$-0.353610\pi$
$439$	18.5997	0.887715	0.443858	+	0.896097i	$0.353610\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	17.1084	0.812842	0.406421	−	0.913686i	$-0.366777\pi$
$443$	17.1084	0.812842	0.406421	+	0.913686i	$0.366777\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.53919	0.403890
$448$	0	0
$449$	−20.9083	−0.986723	−0.493362	−	0.869824i	$-0.664232\pi$
$449$	−20.9083	−0.986723	−0.493362	+	0.869824i	$0.664232\pi$
$450$	0	0
$451$	−2.77205	−0.130531
$452$	0	0
$453$	2.63090	0.123610
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.52586	−0.398823	−0.199411	−	0.979916i	$-0.563903\pi$
$457$	−8.52586	−0.398823	−0.199411	+	0.979916i	$0.563903\pi$
$458$	0	0
$459$	−3.24846	−0.151625
$460$	0	0
$461$	−8.73820	−0.406979	−0.203489	−	0.979077i	$-0.565228\pi$
$461$	−8.73820	−0.406979	−0.203489	+	0.979077i	$0.565228\pi$
$462$	0	0
$463$	−33.8960	−1.57528	−0.787640	−	0.616135i	$-0.788698\pi$
$463$	−33.8960	−1.57528	−0.787640	+	0.616135i	$0.788698\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.00946	−0.0929865	−0.0464933	−	0.998919i	$-0.514805\pi$
$467$	−2.00946	−0.0929865	−0.0464933	+	0.998919i	$0.514805\pi$
$468$	0	0
$469$	8.31124	0.383778
$470$	0	0
$471$	−16.7587	−0.772201
$472$	0	0
$473$	2.68035	0.123242
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.80098	−0.219822
$478$	0	0
$479$	10.5392	0.481548	0.240774	−	0.970581i	$-0.422599\pi$
$479$	10.5392	0.481548	0.240774	+	0.970581i	$0.422599\pi$
$480$	0	0
$481$	−15.8504	−0.722718
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.09171	0.366670	0.183335	−	0.983050i	$-0.441311\pi$
$487$	8.09171	0.366670	0.183335	+	0.983050i	$0.441311\pi$
$488$	0	0
$489$	5.26180	0.237947
$490$	0	0
$491$	−26.9455	−1.21603	−0.608016	−	0.793925i	$-0.708034\pi$
$491$	−26.9455	−1.21603	−0.608016	+	0.793925i	$0.708034\pi$
$492$	0	0
$493$	−27.6453	−1.24508
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.2979	−0.506781
$498$	0	0
$499$	−25.8554	−1.15744	−0.578722	−	0.815525i	$-0.696448\pi$
$499$	−25.8554	−1.15744	−0.578722	+	0.815525i	$0.696448\pi$
$500$	0	0
$501$	19.4752	0.870087
$502$	0	0
$503$	−6.08557	−0.271342	−0.135671	−	0.990754i	$-0.543319\pi$
$503$	−6.08557	−0.271342	−0.135671	+	0.990754i	$0.543319\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.94441	0.352824
$508$	0	0
$509$	−29.3835	−1.30240	−0.651200	−	0.758906i	$-0.725734\pi$
$509$	−29.3835	−1.30240	−0.651200	+	0.758906i	$0.725734\pi$
$510$	0	0
$511$	−8.09171	−0.357956
$512$	0	0
$513$	−2.24846	−0.0992721
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.78765	−0.210561
$518$	0	0
$519$	−20.4391	−0.897176
$520$	0	0
$521$	22.6647	0.992961	0.496480	−	0.868048i	$-0.334625\pi$
$521$	22.6647	0.992961	0.496480	+	0.868048i	$0.334625\pi$
$522$	0	0
$523$	−37.5441	−1.64169	−0.820845	−	0.571152i	$-0.806497\pi$
$523$	−37.5441	−1.64169	−0.820845	+	0.571152i	$0.806497\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.7948	−0.949398
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.170086	−0.00738112
$532$	0	0
$533$	−11.5597	−0.500707
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.78765	−0.120296
$538$	0	0
$539$	3.23513	0.139347
$540$	0	0
$541$	−28.2329	−1.21383	−0.606913	−	0.794768i	$-0.707592\pi$
$541$	−28.2329	−1.21383	−0.606913	+	0.794768i	$0.707592\pi$
$542$	0	0
$543$	15.9155	0.682999
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.4534	1.38761	0.693805	−	0.720163i	$-0.255933\pi$
$547$	32.4534	1.38761	0.693805	+	0.720163i	$0.255933\pi$
$548$	0	0
$549$	−2.67316	−0.114088
$550$	0	0
$551$	−19.1350	−0.815179
$552$	0	0
$553$	5.57531	0.237086
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.4341	−0.865823	−0.432911	−	0.901437i	$-0.642514\pi$
$557$	−20.4341	−0.865823	−0.432911	+	0.901437i	$0.642514\pi$
$558$	0	0
$559$	11.1773	0.472748
$560$	0	0
$561$	1.75154	0.0739499
$562$	0	0
$563$	−20.0316	−0.844231	−0.422115	−	0.906542i	$-0.638712\pi$
$563$	−20.0316	−0.844231	−0.422115	+	0.906542i	$0.638712\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−43.2762	−1.81423	−0.907116	−	0.420881i	$-0.861721\pi$
$569$	−43.2762	−1.81423	−0.907116	+	0.420881i	$0.861721\pi$
$570$	0	0
$571$	24.0605	1.00690	0.503451	−	0.864024i	$-0.332064\pi$
$571$	24.0605	1.00690	0.503451	+	0.864024i	$0.332064\pi$
$572$	0	0
$573$	1.12064	0.0468153
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−35.9877	−1.49819	−0.749094	−	0.662464i	$-0.769511\pi$
$577$	−35.9877	−1.49819	−0.749094	+	0.662464i	$0.769511\pi$
$578$	0	0
$579$	−12.8638	−0.534600
$580$	0	0
$581$	0.411363	0.0170662
$582$	0	0
$583$	2.58864	0.107210
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.2618	0.464824	0.232412	−	0.972617i	$-0.425338\pi$
$587$	11.2618	0.464824	0.232412	+	0.972617i	$0.425338\pi$
$588$	0	0
$589$	−15.0856	−0.621590
$590$	0	0
$591$	11.7854	0.484786
$592$	0	0
$593$	−1.56424	−0.0642357	−0.0321179	−	0.999484i	$-0.510225\pi$
$593$	−1.56424	−0.0642357	−0.0321179	+	0.999484i	$0.510225\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.1194	1.02807
$598$	0	0
$599$	−12.2290	−0.499663	−0.249831	−	0.968289i	$-0.580375\pi$
$599$	−12.2290	−0.499663	−0.249831	+	0.968289i	$0.580375\pi$
$600$	0	0
$601$	24.6658	1.00614	0.503069	−	0.864246i	$-0.332204\pi$
$601$	24.6658	1.00614	0.503069	+	0.864246i	$0.332204\pi$
$602$	0	0
$603$	8.31124	0.338460
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.5574	−0.834401	−0.417200	−	0.908815i	$-0.636989\pi$
$607$	−20.5574	−0.834401	−0.417200	+	0.908815i	$0.636989\pi$
$608$	0	0
$609$	8.51026	0.344853
$610$	0	0
$611$	−19.9649	−0.807695
$612$	0	0
$613$	−40.7526	−1.64598	−0.822991	−	0.568055i	$-0.807696\pi$
$613$	−40.7526	−1.64598	−0.822991	+	0.568055i	$0.807696\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.7019	1.27627	0.638135	−	0.769924i	$-0.279706\pi$
$617$	31.7019	1.27627	0.638135	+	0.769924i	$0.279706\pi$
$618$	0	0
$619$	40.5692	1.63061	0.815306	−	0.579030i	$-0.196569\pi$
$619$	40.5692	1.63061	0.815306	+	0.579030i	$0.196569\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−6.68035	−0.267642
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.21235	0.0484165
$628$	0	0
$629$	22.8999	0.913078
$630$	0	0
$631$	22.6297	0.900873	0.450437	−	0.892808i	$-0.351268\pi$
$631$	22.6297	0.900873	0.450437	+	0.892808i	$0.351268\pi$
$632$	0	0
$633$	−4.51745	−0.179552
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.4908	0.534524
$638$	0	0
$639$	−11.2979	−0.446939
$640$	0	0
$641$	8.50799	0.336045	0.168023	−	0.985783i	$-0.446262\pi$
$641$	8.50799	0.336045	0.168023	+	0.985783i	$0.446262\pi$
$642$	0	0
$643$	−25.1795	−0.992984	−0.496492	−	0.868041i	$-0.665379\pi$
$643$	−25.1795	−0.992984	−0.496492	+	0.868041i	$0.665379\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.53919	0.178454	0.0892270	−	0.996011i	$-0.471560\pi$
$647$	4.53919	0.178454	0.0892270	+	0.996011i	$0.471560\pi$
$648$	0	0
$649$	0.0917087	0.00359988
$650$	0	0
$651$	6.70928	0.262957
$652$	0	0
$653$	−11.1038	−0.434526	−0.217263	−	0.976113i	$-0.569713\pi$
$653$	−11.1038	−0.434526	−0.217263	+	0.976113i	$0.569713\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.09171	−0.315688
$658$	0	0
$659$	18.2713	0.711747	0.355873	−	0.934534i	$-0.384183\pi$
$659$	18.2713	0.711747	0.355873	+	0.934534i	$0.384183\pi$
$660$	0	0
$661$	31.2306	1.21473	0.607365	−	0.794423i	$-0.292227\pi$
$661$	31.2306	1.21473	0.607365	+	0.794423i	$0.292227\pi$
$662$	0	0
$663$	7.30406	0.283666
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.51026	0.329519
$668$	0	0
$669$	3.31965	0.128345
$670$	0	0
$671$	1.44134	0.0556422
$672$	0	0
$673$	−31.3535	−1.20859	−0.604294	−	0.796761i	$-0.706545\pi$
$673$	−31.3535	−1.20859	−0.604294	+	0.796761i	$0.706545\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.8215	1.06927	0.534634	−	0.845084i	$-0.320450\pi$
$677$	27.8215	1.06927	0.534634	+	0.845084i	$0.320450\pi$
$678$	0	0
$679$	−14.8865	−0.571293
$680$	0	0
$681$	−19.5174	−0.747910
$682$	0	0
$683$	−38.2544	−1.46377	−0.731883	−	0.681431i	$-0.761358\pi$
$683$	−38.2544	−1.46377	−0.731883	+	0.681431i	$0.761358\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.39189	0.358323
$688$	0	0
$689$	10.7948	0.411251
$690$	0	0
$691$	13.9506	0.530704	0.265352	−	0.964152i	$-0.414512\pi$
$691$	13.9506	0.530704	0.265352	+	0.964152i	$0.414512\pi$
$692$	0	0
$693$	−0.539189	−0.0204821
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.7009	0.632590
$698$	0	0
$699$	15.1506	0.573049
$700$	0	0
$701$	−20.1678	−0.761728	−0.380864	−	0.924631i	$-0.624373\pi$
$701$	−20.1678	−0.761728	−0.380864	+	0.924631i	$0.624373\pi$
$702$	0	0
$703$	15.8504	0.597810
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.12064	−0.0797548
$708$	0	0
$709$	15.3724	0.577323	0.288662	−	0.957431i	$-0.406790\pi$
$709$	15.3724	0.577323	0.288662	+	0.957431i	$0.406790\pi$
$710$	0	0
$711$	5.57531	0.209090
$712$	0	0
$713$	6.70928	0.251264
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.5536	1.02901
$718$	0	0
$719$	38.2378	1.42603	0.713014	−	0.701149i	$-0.247329\pi$
$719$	38.2378	1.42603	0.713014	+	0.701149i	$0.247329\pi$
$720$	0	0
$721$	−0.290725	−0.0108272
$722$	0	0
$723$	18.6153	0.692310
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.8888	−0.848899	−0.424450	−	0.905452i	$-0.639532\pi$
$727$	−22.8888	−0.848899	−0.424450	+	0.905452i	$0.639532\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.1483	−0.597268
$732$	0	0
$733$	−3.00841	−0.111118	−0.0555591	−	0.998455i	$-0.517694\pi$
$733$	−3.00841	−0.111118	−0.0555591	+	0.998455i	$0.517694\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.48133	−0.165072
$738$	0	0
$739$	−8.15449	−0.299968	−0.149984	−	0.988688i	$-0.547922\pi$
$739$	−8.15449	−0.299968	−0.149984	+	0.988688i	$0.547922\pi$
$740$	0	0
$741$	5.05559	0.185722
$742$	0	0
$743$	42.8827	1.57321	0.786606	−	0.617455i	$-0.211836\pi$
$743$	42.8827	1.57321	0.786606	+	0.617455i	$0.211836\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.411363	0.0150510
$748$	0	0
$749$	12.3763	0.452220
$750$	0	0
$751$	49.9637	1.82320	0.911601	−	0.411077i	$-0.134847\pi$
$751$	49.9637	1.82320	0.911601	+	0.411077i	$0.134847\pi$
$752$	0	0
$753$	4.18342	0.152452
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.62863	0.313613	0.156806	−	0.987629i	$-0.449880\pi$
$757$	8.62863	0.313613	0.156806	+	0.987629i	$0.449880\pi$
$758$	0	0
$759$	−0.539189	−0.0195713
$760$	0	0
$761$	−37.3617	−1.35436	−0.677181	−	0.735817i	$-0.736799\pi$
$761$	−37.3617	−1.35436	−0.677181	+	0.735817i	$0.736799\pi$
$762$	0	0
$763$	12.1906	0.441330
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.382433	0.0138089
$768$	0	0
$769$	22.4775	0.810558	0.405279	−	0.914193i	$-0.367174\pi$
$769$	22.4775	0.810558	0.405279	+	0.914193i	$0.367174\pi$
$770$	0	0
$771$	12.9060	0.464799
$772$	0	0
$773$	8.28685	0.298057	0.149029	−	0.988833i	$-0.452385\pi$
$773$	8.28685	0.298057	0.149029	+	0.988833i	$0.452385\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.04945	−0.252898
$778$	0	0
$779$	11.5597	0.414170
$780$	0	0
$781$	6.09171	0.217978
$782$	0	0
$783$	8.51026	0.304132
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.1734	0.719104	0.359552	−	0.933125i	$-0.382930\pi$
$787$	20.1734	0.719104	0.359552	+	0.933125i	$0.382930\pi$
$788$	0	0
$789$	−4.70209	−0.167399
$790$	0	0
$791$	1.72261	0.0612488
$792$	0	0
$793$	6.01050	0.213439
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.832181	0.0294774	0.0147387	−	0.999891i	$-0.495308\pi$
$797$	0.832181	0.0294774	0.0147387	+	0.999891i	$0.495308\pi$
$798$	0	0
$799$	28.8443	1.02044
$800$	0	0
$801$	−6.68035	−0.236038
$802$	0	0
$803$	4.36296	0.153965
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.5246	1.07452
$808$	0	0
$809$	−9.01106	−0.316812	−0.158406	−	0.987374i	$-0.550636\pi$
$809$	−9.01106	−0.316812	−0.158406	+	0.987374i	$0.550636\pi$
$810$	0	0
$811$	−50.7152	−1.78085	−0.890426	−	0.455127i	$-0.849594\pi$
$811$	−50.7152	−1.78085	−0.890426	+	0.455127i	$0.849594\pi$
$812$	0	0
$813$	2.65983	0.0932842
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.1773	−0.391043
$818$	0	0
$819$	−2.24846	−0.0785677
$820$	0	0
$821$	16.0761	0.561060	0.280530	−	0.959845i	$-0.409490\pi$
$821$	16.0761	0.561060	0.280530	+	0.959845i	$0.409490\pi$
$822$	0	0
$823$	−39.0928	−1.36269	−0.681344	−	0.731963i	$-0.738604\pi$
$823$	−39.0928	−1.36269	−0.681344	+	0.731963i	$0.738604\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.8732	−0.586739	−0.293370	−	0.955999i	$-0.594777\pi$
$827$	−16.8732	−0.586739	−0.293370	+	0.955999i	$0.594777\pi$
$828$	0	0
$829$	−36.8492	−1.27983	−0.639913	−	0.768447i	$-0.721030\pi$
$829$	−36.8492	−1.27983	−0.639913	+	0.768447i	$0.721030\pi$
$830$	0	0
$831$	−0.581449	−0.0201703
$832$	0	0
$833$	−19.4908	−0.675316
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.70928	0.231906
$838$	0	0
$839$	6.70701	0.231552	0.115776	−	0.993275i	$-0.463065\pi$
$839$	6.70701	0.231552	0.115776	+	0.993275i	$0.463065\pi$
$840$	0	0
$841$	43.4245	1.49740
$842$	0	0
$843$	−29.3907	−1.01227
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.7093	−0.367975
$848$	0	0
$849$	−2.23513	−0.0767096
$850$	0	0
$851$	−7.04945	−0.241652
$852$	0	0
$853$	4.96719	0.170074	0.0850368	−	0.996378i	$-0.472899\pi$
$853$	4.96719	0.170074	0.0850368	+	0.996378i	$0.472899\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.3751	1.00343	0.501717	−	0.865032i	$-0.332702\pi$
$857$	29.3751	1.00343	0.501717	+	0.865032i	$0.332702\pi$
$858$	0	0
$859$	−36.8720	−1.25806	−0.629028	−	0.777383i	$-0.716547\pi$
$859$	−36.8720	−1.25806	−0.629028	+	0.777383i	$0.716547\pi$
$860$	0	0
$861$	−5.14116	−0.175210
$862$	0	0
$863$	22.7838	0.775569	0.387784	−	0.921750i	$-0.373241\pi$
$863$	22.7838	0.775569	0.387784	+	0.921750i	$0.373241\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.44748	0.218968
$868$	0	0
$869$	−3.00614	−0.101976
$870$	0	0
$871$	−18.6875	−0.633203
$872$	0	0
$873$	−14.8865	−0.503833
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.34471	0.281781	0.140890	−	0.990025i	$-0.455003\pi$
$877$	8.34471	0.281781	0.140890	+	0.990025i	$0.455003\pi$
$878$	0	0
$879$	−6.01333	−0.202825
$880$	0	0
$881$	−26.9926	−0.909405	−0.454702	−	0.890643i	$-0.650254\pi$
$881$	−26.9926	−0.909405	−0.454702	+	0.890643i	$0.650254\pi$
$882$	0	0
$883$	6.30179	0.212072	0.106036	−	0.994362i	$-0.466184\pi$
$883$	6.30179	0.212072	0.106036	+	0.994362i	$0.466184\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.82273	0.161931	0.0809656	−	0.996717i	$-0.474200\pi$
$887$	4.82273	0.161931	0.0809656	+	0.996717i	$0.474200\pi$
$888$	0	0
$889$	−12.1639	−0.407966
$890$	0	0
$891$	−0.539189	−0.0180635
$892$	0	0
$893$	19.9649	0.668101
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.24846	−0.0750740
$898$	0	0
$899$	57.0977	1.90431
$900$	0	0
$901$	−15.5958	−0.519572
$902$	0	0
$903$	4.97107	0.165427
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.4163	0.379071	0.189536	−	0.981874i	$-0.439302\pi$
$907$	11.4163	0.379071	0.189536	+	0.981874i	$0.439302\pi$
$908$	0	0
$909$	−2.12064	−0.0703371
$910$	0	0
$911$	−42.1933	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$911$	−42.1933	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$912$	0	0
$913$	−0.221802	−0.00734058
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.78539	0.257096
$918$	0	0
$919$	−15.7731	−0.520307	−0.260153	−	0.965567i	$-0.583773\pi$
$919$	−15.7731	−0.520307	−0.260153	+	0.965567i	$0.583773\pi$
$920$	0	0
$921$	0.516403	0.0170160
$922$	0	0
$923$	25.4030	0.836148
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.290725	−0.00954865
$928$	0	0
$929$	−20.0133	−0.656616	−0.328308	−	0.944571i	$-0.606478\pi$
$929$	−20.0133	−0.656616	−0.328308	+	0.944571i	$0.606478\pi$
$930$	0	0
$931$	−13.4908	−0.442143
$932$	0	0
$933$	11.5031	0.376594
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.0433	0.687455	0.343727	−	0.939069i	$-0.388310\pi$
$937$	21.0433	0.687455	0.343727	+	0.939069i	$0.388310\pi$
$938$	0	0
$939$	−3.70313	−0.120847
$940$	0	0
$941$	−24.9204	−0.812382	−0.406191	−	0.913788i	$-0.633143\pi$
$941$	−24.9204	−0.812382	−0.406191	+	0.913788i	$0.633143\pi$
$942$	0	0
$943$	−5.14116	−0.167419
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.9421	1.29795	0.648973	−	0.760812i	$-0.275199\pi$
$947$	39.9421	1.29795	0.648973	+	0.760812i	$0.275199\pi$
$948$	0	0
$949$	18.1939	0.590600
$950$	0	0
$951$	18.9204	0.613536
$952$	0	0
$953$	25.8310	0.836747	0.418373	−	0.908275i	$-0.362600\pi$
$953$	25.8310	0.836747	0.418373	+	0.908275i	$0.362600\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.58864	−0.148330
$958$	0	0
$959$	9.86603	0.318591
$960$	0	0
$961$	14.0144	0.452077
$962$	0	0
$963$	12.3763	0.398820
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.3802	−0.558908	−0.279454	−	0.960159i	$-0.590154\pi$
$967$	−17.3802	−0.558908	−0.279454	+	0.960159i	$0.590154\pi$
$968$	0	0
$969$	−7.30406	−0.234640
$970$	0	0
$971$	61.4908	1.97333	0.986667	−	0.162754i	$-0.0520378\pi$
$971$	61.4908	1.97333	0.986667	+	0.162754i	$0.0520378\pi$
$972$	0	0
$973$	4.07838	0.130747
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.2940	−0.361328	−0.180664	−	0.983545i	$-0.557825\pi$
$977$	−11.2940	−0.361328	−0.180664	+	0.983545i	$0.557825\pi$
$978$	0	0
$979$	3.60197	0.115119
$980$	0	0
$981$	12.1906	0.389216
$982$	0	0
$983$	34.0010	1.08446	0.542232	−	0.840229i	$-0.317579\pi$
$983$	34.0010	1.08446	0.542232	+	0.840229i	$0.317579\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.87936	−0.282633
$988$	0	0
$989$	4.97107	0.158071
$990$	0	0
$991$	35.2023	1.11824	0.559119	−	0.829087i	$-0.311139\pi$
$991$	35.2023	1.11824	0.559119	+	0.829087i	$0.311139\pi$
$992$	0	0
$993$	−1.29072	−0.0409599
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.91548	−0.0606638	−0.0303319	−	0.999540i	$-0.509656\pi$
$997$	−1.91548	−0.0606638	−0.0303319	+	0.999540i	$0.509656\pi$
$998$	0	0
$999$	−7.04945	−0.223035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.y.1.2		3
5.2	odd	4		1380.2.f.a.829.5	yes	6
5.3	odd	4		1380.2.f.a.829.2	✓	6
5.4	even	2		6900.2.a.z.1.2		3
15.2	even	4		4140.2.f.a.829.4		6
15.8	even	4		4140.2.f.a.829.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.a.829.2	✓	6	5.3	odd	4
1380.2.f.a.829.5	yes	6	5.2	odd	4
4140.2.f.a.829.3		6	15.8	even	4
4140.2.f.a.829.4		6	15.2	even	4
6900.2.a.y.1.2		3	1.1	even	1	trivial
6900.2.a.z.1.2		3	5.4	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -0.539189 q^{11} -2.24846 q^{13} +3.24846 q^{17} +2.24846 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{27} -8.51026 q^{29} -6.70928 q^{31} +0.539189 q^{33} +7.04945 q^{37} +2.24846 q^{39} +5.14116 q^{41} -4.97107 q^{43} +8.87936 q^{47} -6.00000 q^{49} -3.24846 q^{51} -4.80098 q^{53} -2.24846 q^{57} -0.170086 q^{59} -2.67316 q^{61} +1.00000 q^{63} +8.31124 q^{67} +1.00000 q^{69} -11.2979 q^{71} -8.09171 q^{73} -0.539189 q^{77} +5.57531 q^{79} +1.00000 q^{81} +0.411363 q^{83} +8.51026 q^{87} -6.68035 q^{89} -2.24846 q^{91} +6.70928 q^{93} -14.8865 q^{97} -0.539189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{13} + q^{17} - 2 q^{19} - 3 q^{21} - 3 q^{23} - 3 q^{27} - 9 q^{29} - 13 q^{31} + 3 q^{37} - 2 q^{39} - 5 q^{41} + 14 q^{47} - 18 q^{49} - q^{51} - 5 q^{53} + 2 q^{57} + 5 q^{59} - 20 q^{61} + 3 q^{63} - q^{67} + 3 q^{69} - 7 q^{71} - 22 q^{73} - 4 q^{79} + 3 q^{81} + 21 q^{83} + 9 q^{87} + 2 q^{89} + 2 q^{91} + 13 q^{93} + 2 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^13 + q^17 - 2 * q^19 - 3 * q^21 - 3 * q^23 - 3 * q^27 - 9 * q^29 - 13 * q^31 + 3 * q^37 - 2 * q^39 - 5 * q^41 + 14 * q^47 - 18 * q^49 - q^51 - 5 * q^53 + 2 * q^57 + 5 * q^59 - 20 * q^61 + 3 * q^63 - q^67 + 3 * q^69 - 7 * q^71 - 22 * q^73 - 4 * q^79 + 3 * q^81 + 21 * q^83 + 9 * q^87 + 2 * q^89 + 2 * q^91 + 13 * q^93 + 2 * q^97

Introduction

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